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Noncommutative Prüfer rings and some generalizations Zhou, Yiqiang 1993

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NONCOMMUTATIVE PRUFER RINGS AND SOME GENERALIZATIONS by YIQIANG ZHOU B.Sc., Hunan Normal University, 1981 M.Sc., Beijing Normal University, 1984 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department of Mathematics) We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA April 1993 ©Yiqiang Zhou, 1993  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  (Signature)  Department of ^Mathematics The University of British Columbia Vancouver, Canada  Date ^April 30. I 993  DE-6 (2/88)  11  Abstract Noncommutative Prfifer rings appear naturally when one wants to transfer the known results for rings which arise in algebraic geometry (such as Dedekind, Krull and Priifer, valuation rings ...) to noncommutative rings. We remove the left-right symmetry condition of the noncommutative Prfifer rings introduced by Alajbegovic and Dubrovin, and introduce three natural generalizations, semi-Prfifer rings, right w-semi-Prfifer rings, and right wPrfifer rings. We study the relations between the four concepts, and present the various properties that characterize them. We formulate and prove the basic facts for those rings (decompositions of such rings; Morita invariants of these notions; relations with some other notions). A new module-theoretic characterization of semiprime right Goldie rings is achieved by using the newly-defined concept of strongly compressible modules. The result is used to provide new characterizations of semiprime Goldie (prime right Goldie, or prime Goldie) rings, and right w-semi-Prfifer (semi-Prfifer, right w-Prfifer, or Prfifer) rings. In particular, the characterization of semiprime Goldie rings of Lopez-Permouth, Rizvi, and Yousif using weakly-injective modules is an easy corollary of our results. We also study modules over noncommutative Priifer rings. It is shown that a module over a noncommutative Prfifer ring has projective dimension at most one if and only if it is the union of a well-ordered continuous chain of submodules with each factor of the chain a finitely presented cyclic module. The result is used to present a characterization of divisible modules with projective dimension at most one over noncommutative Priifer rings, which generalizes a known result of L.Fuchs.  iii Contents Abstract^  ii  Table of contents^  iii  Notations^  iv  Acknowledgements^  vii  Introduction^  1  1 The Preliminaries^  1^p  2 Noncommutative Priifer rings and some generalizations ^10  2.1 Definitions and properties  ^11  2.2 A structure theorem and further properties of right w-semi-Priifer rings  2.3 Priifer rings and semi-Priifer rings 3 Strongly compressible modules^  ^21 ^27 40  3.1 New characterizations of semiprime right Goldie rings 41 3.2 Some applications 4 Modules over Priifer rings ^  4.1 Modules of projective dimension at most one ^  ^47 54  54  4.2 Divisible modules of projective dimension at most one 67 References^  79  iv  Notations Notations in this manuscript are fairly standard, and may be found in most graduate level texts on Algebra and Ring Theory. To keep the reader on track, we will introduce them as required. The following two books are our main references:  1) Rings and Categories of Modules by F.W. Anderson and K.R. Fuller, and 2) An Introduction to Noncommutative Noetherian Rings by K.R. Goodearl and R.B. Warfield. Jr. We will feel free to use the results in the two books whenever we have such a demand. Throughout this manuscript, a ring R will mean a nonzero associative noncommutative ring with an identity. And all modules are unitary. The notation MR (or  R M)  indicates that M is a right (or left) module over a  ring R. Given a module MR, we will denote by E(MR ) the injective hull of the module MR. For a subset X of a right R-module M, the annihilator right ideal of X in R is denoted by X', i.e., X' = {r E R : xr = 0 for all  x E X}. Similarly, for a left R-module  RN  and a subset Y of N, we denote  the annihilator left ideal of Y in R by 1 Y. In particular, we write x -l- (or 'y) to indicate {x} -1- (or '{y}). Mod-R^the category of all right R-modules R-Mod^the category of all left R-modules C^proper inclusion  Notations ^ N^the set of positive integers Z^the set of integers End(M)^the ring of all module endomorphisms of a module M  Z(MR )^the singular submodule of a module MR T(M)^the torsion submodule of a module M  T(M)^the trace ideal of a module M M*^the dual module of a module M dim(M)^the Goldie dimension of a module M Pd(M)^the projective dimension of a module M mu)^the direct sum of I copies of M  M(n)^the direct sum of n copies of M Mn (R)^the n by n matrix ring over a ring R Rad(R)^the Jacobson radical of a ring R CR(0)^the set of all regular elements of R Qrd(R)^the classical right quotient ring of a ring R (if it exists) Q ict(R)^the classical left quotient ring of a ring R (if it exists) Q c /(R)^the classical quotient ring of a ring R (if it exists)  v  Notations^  ACC^the ascending chain condition 0^tensor product Ext^the extension functor  vi  vii  Acknowledgements I would like to express my deep thanks to my thesis supervisor, Dr. Stanley S.Page, for his invaluable advice and kindly encouragement throughout past few years. The many many hours of discussion with him on ring theory made my stay at the University of British Columbia a most pleasurable experience of study of my life. I would especially like to thank my wife, Hongwa, for her support, love, and understanding.  1  Introduction Priifer domains form an important and much-studied class of integral domains in Commutative Algebra. With Dedekind domains, valuation domains and Krull domains, they constitute the main objects of study in the Multiplicative Theory of Ideals. The importance of the class of Priifer domains lies mainly in: 1) Priifer domains have an origin in Algebraic Number Theory. The rings of integers of finite algebraic number fields, which are the main objects of study in Algebraic Number Theory, are Priifer domains. 2) Priifer domains have tight connections with Dedekind domains and valuation domains. In fact, the class of Dedekind domains is precisely the class of Noetherian Priifer domains; and a Priifer domain can be characterized as an integral domain such that the localization of it at any prime (or maximal) ideal is a valuation domain. 3) The lattice of all ideals of a Priifer domain possesses many beautiful arithmetics. For example, an integral domain is a Priifer domain if and only if A(B fl C) = AB fl AC for all ideals A, B, C of  R if and only if A fl (B + C) = An B + A fl C for all ideals A, B, C of R. In the past twenty years, the study of the noncommutative analogues of Dedekind domains, valuation domains and Krull domains has been a fascinating area of study in ring theory. And many results have been obtained on the various generalizations of them to noncommutative cases, e.g., Asano orders, Dedekind prime rings, hereditary Noetherian prime rings, chain rings, Dubrovin valuation rings, Chamerie Krull rings, Marubaynshi Krull rings, flKrull rings, and others. It is from the abundance of the study of these objects and the close relations between Priifer domains, Dedekind domains, valua-  Introduction  ^  IA  tion domains and Krull domains that one sees the need for an introduction of noncommutative analogues of Priifer domains. Let R be an integral domain with field of quotients Q. An R-submodule  I of Q is said to be a fractional ideal of R if dI C R for some 0 d E R. For each fractional ideal I of R, define I  -  = {q E Q : qI C R}. Then I is said  to be invertible if I I = R. We call an integral domain R a Priifer domain -  if every nonzero finitely generated (f.g. for short) ideal of R is invertible. Priifer domains can be characterized as any commutative rings with the property that each nonzero f.g. ideal is a progenerator, or is projective, or is a generator. This large selection of attributes suggests many possible generalizations to noncommutative cases, and at the same time raises the difficulty of the best choice among such numerous generalizations. In 1990, Alajbegovic and Dubrovin defined a noncommutative (right) Priifer ring as a prime Goldie ring such that / -1 / = R and II' = 00) for every f.g. fractional right ideal I of R, where 0/(/) = {q E Q c /(/) : qI C I} and /' {q E Qd(R) : IqI  c  I}. Among the properties of noncommutative  Priifer rings, they show that the concept of a noncommutative Priifer ring is a left-right symmetric concept; the notion is a Morita invariant, and every noncommutative Priifer ring is Morita equivalent to a (noncommutative) Priifer domain. They also note that the class of noncommutative Priifer rings contains the classes of prime Dedekind rings, Dubrovin valuation rings, and commutative Priifer domains. The present manuscript is devoted to continuing the study of noncommutative Priifer rings. We first observe that a noncommutative Priifer ring can be characterized as a prime Goldie ring R such that every nonzero f.g.  Introduction submodule of a progenerator of Mod-R is a progenerator. The nature of the characterization brought our interests to noncommutative Prfifer rings. We note that the generalized discrete valuation ring of H.H.Brungs (see [5]) and the skew polynomial rings (see the example in §2.1, of Chapter 2) provide examples of prime right (but not left) Goldie rings satisfying the same property as above. The observation leads us to remove the left-right symmetric condition of noncommutative Prfifer rings and to consider more general definitions where the conditions of being a prime ring, being a Goldie ring are replaced by a semiprime ring, by a right Goldie ring respectively. The manuscript is organized into four chapters. Chapter 1 summarizes certain basic concepts and theorems in ring theory which are needed in the sequel. Since they are all well-known and easy to find for reference, the proofs of most of them are omitted. In Chapter 2, we introduce three generalizations of noncommutative Prfifer rings, semi-Prfifer rings, right w-semi-Prfifer rings, and right w-Priifer rings. We study the relations between the four concepts, and present the various properties that characterize them. We formulate and prove the basic facts for those rings (decompositions of such rings; Morita invariants of these notions; relations with some other notions). In Chapter 3, a new module-theoretic characterization of semiprime right Goldie rings is achieved by using the newly-defined concept of strongly compressible modules. The result is used to provide new characterizations of semiprime Goldie (prime right Goldie, or prime Goldie) rings, and right wsemi-Prfifer (semi-Prfifer, right w-Prfifer, or Prfifer) rings. In particular, the characterization of semiprime Goldie rings of Lopez-Permouth, Rizvi, and  Introduction^  1C  Yousif using weakly-injective modules is an easy corollary of our results. Chapter 4 is provided to study modules over noncommutative Priifer rings. The study is motivated by the work of L.Fuchs on modules over valuation (or Priifer) domains (see [13]). We give a characterization of modules of projective dimension at most one over noncommutative Priifer rings, and present a structure theorem of divisible modules with projective dimension at most one over noncommutative Priifer rings, which generalizes a known result of L.Fuchs.  it)  1 The Preliminaries This chapter is provided to review a number of basic concepts and some important results from ring theory, which will be used throughout the sequel. The proofs for most results are omitted, since they can be found in the standard texts in ring theory, such as [2] and [18]. Essential extensions and singular submodules An essential submodule of a module M is any submodule N which has nonzero intersection with every nonzero submodule of M. We write N < e M to denote this situation, and we also say that M is an essential extension of N. Proposition 1.1 (a) Let N be a submodule of a module M, and let f : P —÷ M be a homomorphism. If N <, M, then f -1 (N) <, P. (b) Let N be a submodule of a module M, and P a submodule of M which is maximal with respect to the property P fl N = 0. Then N f P <, M and (N E P)/P  < e M/P.  ^  The singular submodule of a module MR is defined by Z(MR ) = {x E M : <, RR}. Since Z(MR ) is a fully invariant submodule of M, the right singular ideal Z(R R ) is an ideal of R. If Z(MR ) = 0 then M is called a nonsingular module. The ring R is called a right non-singular ring if Z(RR) = 0. A right and left non-singular ring is called a non-singular ring.  The Preliminaries  ^  2  Orders and quotient rings  A regular element in a ring R is any non-zero-divisor, i.e., any element x E R such that xi = lx --= 0. We will denote by CR(0) the set of all regular  elements of R. Definition 1.1 Let Q be a ring. A right order in Q is any sabring R C Q  such that (a) every regular element of R is invertible in Q; (b) every element of Q has the form ab -1 for some a E R and some b E CR(0).  A left order is defined analogously, and a left and right order is called an order. Definition 1.2 Let R be a ring. A classical right quotient ring, denoted by  Q cri (R) if it exists, is any overring Q D R such that R is a right order in Q. A classical left quotient ring is defined analogously, and a classical left and right quotient ring is called a classical quotient ring.  In Asano [3] it is shown that Qrd (R) exists if and only if R satisfies the right Ore condition, i.e., for any a E R and any c E CR(0) there exist b E R and d E CR(0) such that ad = cb (a right (or left) Ore ring is any ring satisfying  the right (or left) Ore condition). When both Qrd (R) and Q ici (R) exist, we have Q cri (R) 2=2 (2,1 / (R). This occurs only when R is an order. We will denote by IQ d(R) the classical quotient ring of R (if it exists). Another basic fact is that the classical right quotient ring (if it exists) is unique, up to isomorphism (see [18, Cor.9.5, P146]).  The Preliminaries^  3  Lemma 1.1 Let R be a right order with Q = Qra (R) and let S be an overring of R, i.e., R C S C Q. If I is a right S-submodule of Q such that I contains a regular element of R, then Homs(Is, Ss) = {o : q E Q, qI C S}, where o- 9 : I —.-. S is a S-homomorphism defined by a q (x) = qx.  Proof. For each q E Q with qI C S, it is easy to see that o- q is a Shomomorphism. Suppose 0: I --> S is a S-homomorphism. Let s E I be  a regular element of R. For each x E I there exists a regular element t of R such that xt E R. Now, by the right Ore condition, there exist a E R  and u E CR(0) such that sa = xtu. Then 0(x)tu = cb(xtu) = cb(sa) = cb(s)a = cb(s)s'sa = cb(s)s -i xtu, which implies c¢(x) = cb(s)s -l x = o-q (x)  with q = 0(s).9 -1 satisfying qI C S. ^ Goldie rings and Goldie Theorems  A right annihilator in a ring R is any right ideal I of R such that I = X ± for some X C R. Left annihilators are defined in a similar way. Note that a right ideal I is a right annihilator if and only if I = ( I I) .1) -L1 A module MR is called finite-dimensional (or in other words, MR has finite Goldie dimension) if M does not contain an infinite direct sum of nonzero submodules. In this case, there exists a nonnegative integer n such that M contains a direct sum of n nonzero submodules, but no direct sum of n -I- 1 nonzero submodules. Such an n is uniquely determined by M. We shall call this integer the Goldie dimension of MR, and denote it by dim(MR ). Definition 1.3 A right Goldie ring is any ring R such that RR is finite-  dimensional and R has ACC on right annihilators.  The Preliminaries^  4  Proposition 1.2 [Goldie]. Let R be a semiprime right Goldie ring, and let  I be a right ideal of R. Then I is an essential right ideal if and only if I contains a regular element. ^ Theorem 1.1 [Goldie]. Let R be a ring.  (a) R is a right order in a semi-simple ring if and only if R is a semiprime right Goldie ring; (b) R is a right order in a simple Artinian ring if and only if R is a prime right Goldie ring. ^ Theorem 1.2 Let R be semiprime. Then R is a right Goldie ring if and  only if Z(RR) = 0, and RR is finite-dimensional. ^ Torsion modules and torsionfree modules  Given a module MR, let r(M) {x E M : xr 0 for some r E CR(0)}. If R is a right order, then r(M) is a submodule of M. In fact, for x, y E r(M) and  r E R, we have xs 0 yt for some s, t E CR(0). By the right Ore condition, there exist c, d E CR(0), and a ,b E R, such that se = to and ad = rb. Then we have (x — y)sc xsc — ytd 0 and (xa)d = xrb 0. When T(M) is a submodule, it is called the torsion submodule of M. If T(M) M, then M is called a torsion module, and if r(M) = 0, then M is called a torsionfree module. Clearly Mir(M) is torsionfree for every module MR. If R is a semiprime Goldie ring, then, because of Proposition 1.2, Z(MR) = T(MR) for every module MR.  5  The Preliminaries^  Theorem 1.3 [Gentile, Levy]. If R is a semiprime Goldie ring and M is a  f.g. torsionfree right R-module, then M can be embedded in a f.g. free right R-module. ^ Morita equivalences  Given a right R-module M. We let M* = Hom(M, R). The trace of M, written T(MR), is defined by T(M) = Eff(M)  :  f E M*}. It is clear that  T(M) is an ideal of R. We now call a right R-module X a generator of the category Mod-R if the trace ideal T(M) = R. The concept of generator plays a central role in the study of equivalences between categories of modules. The following proposition gives a number of important characterizations of generators. Proposition 1.3 The following are equivalent for a module X E Mod-R:  (a) X is a generator; (b) For every M E Mod-R, there is an index set I such that M is a homomorphism image of X( 1), where V I ) is the direct sum of I copies of X; (c) There exists an n such that R is a homomorphism image of X(n). ^ A module P is called projective if given an epimorphism p : M -->. N, then any homomorphism f : P ---> N can be factored as f = p o g for some  g : P ---. N. It is well-known that a module PR is projective if and only if P is a direct summand of a free module if and only if any short exact sequence 0 - f M -+ N -* P --f 0 splits. A very useful criterion for projectivity is the following proposition which is often called the "dual basis lemma" for projective modules.  The Preliminaries^  6  Proposition 1.4 A R-module PR is projective if and only if there exist a set {x a : a E I} of elements in P and a set {fa : a E I} of elements in P* =Horn (P, R), such that for any x E P, fa (x) = 0 for all but finite number of the fa , and x E aei x,fot (x)• ^ A module X is a progenerator of Mod-R if and only if X is f.g. projective and X is a generator of Mod-R.  Definition 1.4 Let F, G be functors from Mod-R to Mod-S. We say there is a natural isomorphism from F to G, written F G, if there exists a map that assigns to every module M E Mod-R an isomorphism  Om E  Hom s (F(M), G(M)) such that for any M, N E Mod-R and any f E HomR (M, N) the following diagram:  F(M)^G(M) 1F(f)^1G(f)  F(N)^G(N) is commutative.  Definition 1.5 Two rings R and S are said to be Morita equivalent , written R S, if there exist functors F : Mod-R^Mod-S and G : Mod-S --+ Mod-R such that GF^FG "-=" 1 mod-s. In this case, F is called a Morita equivalence and G an inverse equivalence between Mod-R and Mod-S. Any two rings which are isomorphic are of course Morita equivalent. The fact that R  ti  M„,(R), where Mn (R) is the ring of n by n matrices with entries in  R, shows simply that a noncommutative ring may be Morita equivalent to a commutative ring.  The Preliminaries^  7  Theorem 1.4 For two rings R and S, then R^S if and only if S End(M) for some progenerator M of Mod-R. And in this case, Hom R (M, —) : NR^HOMR(S MR, NR) defines a Morita equivalence between Mod-R and Mod-S with inverse equivalence — Os M Ps 1-4 P Os M.  Theorem 1.5 For two rings R and S, then R  ti  ^  S if and only if S L'-  eMn,(R)e for some n and some idempotent e of Mn (R) with M7,(R)eM7,(R) = Mn (R). ^  Any ring property which is preserved under Morita equivalence is called a Morita invariant. For example, being a semiprime right Goldie ring is a Morita invariant because any ring Morita equivalent to a semiprime right Goldie ring is semiprime right Goldie [29, Propo.5.10]. Semihereditary rings A ring is right (or left) semihereditary if every f.g. right (or left) ideal is projective. A right and left semihereditary ring is called a semihereditary ring. An example of a ring which is right but not left semihereditary was given by Chase [6]. In the following, we introduce a theorem of Small which presents certain classes of rings for which right semihereditary implies left semihereditary. Theorem 1.6 [Small]. Let R be a ring in which every principal right ideal is projective and in which there is no infinite set of orthogonal idempotents. Then every right and every left annihilator is generated by an idempotent. In particular, every principal left ideal is projective.  The Preliminaries^  8  Proof. Suppose 0 T S -L . If s E S, then T C si. Thus, T C hR where  h is an idempotent. Now let L be an arbitrary (nonzero) left annihilator. L -L C gR where g 2 = g. But then L = -1- (L -L ) D i (gR) = R(1 — g). Hence, any left annihilator L contains a nontrivial idempotent. By [2, Ex. 10.11], we can choose an idempotent e E L such that -Le is minimal amongst the left annihilators of idempotents in L. We claim  -L e  fl  L = 0. Suppose not. Then  L en L is a nonzero left annihilator which contains an idempotent f 0. Now e* e+ f — ef is an idempotent in L. Since e*e = e, e* 0 and -L e* Cl e. However, fe 0 and fe* = f 0. Thus, -L e* g-Le, which contradicts the minimality of  -L e.  Hence le fl L 0. Now if x E L, then x — xe E L and  (x — xe)e = 0. Therefore x — xe 0 and L = Re. Finally, if K is a right annihilator, then -LK = Re where e 2 = e. But, K = = (1 — e)R. ^ Proposition 1.5 A ring R is right (left) semihereditary if and only if Mn (R),  for all n, has principal right (left) ideals projective. Proof. It is well known that if R is right (left) semihereditary, then so is  111,2 (R). In the other direction, we must show that any f.g. right ideal, say I a 1 R + • • • + a n,R, is projective. In M„,(R) let x be the matrix (c if ) where cii = a, and all other entries are zero. Then xM„(R) is projective as a right  M7,(R)-module. But, x111,2 (R) considered as a right R-module (R embedded in MM (R) in the usual way) is isomorphic to I e••.e I (n times). Thus, since Mn (R) is R-free, Ie•••e/ is R-projective and I is R-projective. ^ Combining Theorem 1.6 and Proposition 1.5, we immediately obtain  The Preliminaries^  9  Theorem 1.7 [Small]. Suppose R is a ring which is right semihereditary  and such that 111,2 (R), for all n, does not possess an infinite set of orthogonal idempotents, then R is left semihereditary. ^  10  2 Noncommutative Priifer rings and some generalizations (Noncommutative) Priifer rings were introduced and studied by Alajbegovic and Dubrovin [1]. Examples of Priifer rings include prime Dedekind rings, commutative Priifer domains and prime Goldie right (or left) Bezout rings (cf.[1]. Examples 1.13 and 1.15). Some important properties of Priifer rings have been demonstrated in the paper of Alajbegovic and Dubrovin (cf.[1] for details). An observation is that a ring R is a Priifer ring if and only if R is a prime Goldie ring with the following property (see Proposition 2.1.1): (P): Every finitely generated essential right ideal of R is a progenerator of Mod-R. Replacing 'prime Goldie' by `semiprime Goldie', 'prime right Goldie', and `semiprime right Goldie', respectively, in the above condition, we introduce three natural generalizations of Priifer ring which are to be called (right) semi-Priifer ring, right w-semi-Priifer ring, and right w-Priifer ring respectively (see section 1 for the precise definitions). The main object of this chapter is to study the relationship between all these rings and to establish various properties of them. In section 1, we first give the definitions of three generalizations of Priifer rings. The four concepts, especially their implication relations, are further explained by using a known example. The rest of section 1 is used to present the various properties and characterizations of all these rings. In section 2, we will present a structure theorem of right w-semi-Priifer rings. " A ring is a right w-semi-Priifer ring if and only if it is a finite direct sum of right  Noncommutative Priifer rings and some generalizations^11  w-Priifer rings". Section 3 is devoted to studying Priifer rings and semiPriifer rings. We will show that the right semi-Priifer rings are exactly the left semi-Priifer rings. A structure theorem states that a ring is a semi-Priifer ring if and only if it is a finite direct sum of Priifer rings. We will pay special attention to the cases where the Priifer ring R is a Noetherian, bounded, semiperfect ring respectively. It was proved in [1] that every Priifer ring is Morita equivalent to a Priifer domain. We will give a stronger result here which says that every Prfifer ring R can be decomposed as a finite direct sum of uniform submodules such that the endomorphism ring of each of these uniform submodules is a Priifer domain which is Morita equivalent to R. The last result can be used to give a characterization of f.g. torsionfree  modules over a semi-Priifer ring.  2.1 Definitions and properties Let ring R be a right order with Q = Qrd (R). Given a subset I of Q, we set O r (I) = {q EQ:IDIO; 01(I) {q E Q : I 2_ qI}; [R :^= {q EQ:RD^[R : = {q EQ:RD qI};  and^I'={EQ:IDIqI}. A submodule I of QR is called a fractional right ideal of R if I contains a regular element of Q, and there exists a regular element d of Q with R D dI. Definition 2.1.1 A semiprime Goldie (semiprime right Goldie or prime right Goldie or prime Goldie) ring R is called a right semi-Pritler (right w-semi-Priifer or right w-Priifer, or right Priifer) ring if every finitely generated (f.g. for short) fractional right ideal I of R satisfies:  Noncommutative Priifer rings and sone generalizations ^12 I' I R,^= 00).  The left-sided versions can be defined in a similar way. Clearly every right Priifer ring is a right semi-Priifer (right w-semi-Priifer or right w-Priifer) ring, and every right w-Priifer ring is a right w-semi-Priifer ring. Remark 2.1.1 The definition of a right Priifer ring is due to Alajbegovic and Dubrovin [1].  Lemma 2.1.1 [32]. If I is a fractional right ideal of a right order R, then the following are equivalent: (a) II -1 = 00); (b) I is a projective right O r (I)-module.  Proof. First we note that given a fractional right ideal I of a right order R, O r (I) is an overring of R and I is a right O r (I)-module.  (a)^(b). By Lemma 1.1, Hom or ( i) (/, O r (/))^{a q : q E Q,qI C O r (I)} {0 q : q E / -1 1, where for each q E  aq  : I Or(I) is the  O r (I)-homomorphism defined by o-q (a) qa. Suppose that / is a projective right O r (I)-module. Then, by the dual basis lemma, there exist {a a : a E X} C I and { o- qa : a E X} C Homo r (/)(/, O r (/)) such that for any a E I, o-qa (a) qa (a) = 0 for all but a finite number of the aq ,,, and a = E a Exa ce g q , JO. Choosing a to be regular shows that q a = 0 for all but a  finite number of a. Letting a be arbitrary again, we see that a E,a,q,a = (E a a a qa )a. Thus E,„a„q a = 1 E //  -  1  and hence // -1 0 / (/).  (a)^(b). Suppose that 0 / (/) = //' Then there exist finite sets {a a } C I and { q a } C / -1 such that E a a„qc, = 1. Hence E a a,q,a = a. Then  Noncommutative Prfifer rings and sone generalizations ^13 E,a a o- q „ (a) = a with each a q „ E Homo r (1)(I,0,(I)). Therefore, by the dual basis lemma, I is a projective right O r (I)-module. ^  Lemma 2.1.2 [1]. If I is a fractional right ideal of a right order R, and L = [R : 1] 1 , then the following are equivalent: (a) LI = R; (b) IR is a generator of Mod-R; (c) I  -  1  I = R.  Any of these conditions implies that O r (I) = R. Proof. (a) <=> (b). By definition, IR is a generator of Mod-R if and only if  R = T(IR) = Eff(I) : f E HomR(/,R)}. By Lemma 1.1, HomR(I , R) = fag : q E L}. Therefore we have that IR is a generator of Mod-R if and only if R= Efa q (/) : q E Ll = Efq/ :q E Ll = LI. Before proving (b) 4=>. (c), we note the fact that if KI R for some subset  K of Q, then O r (I) = R. In fact, for q E Q with Iq C I, we have KIq C KI, i.e., Rq C R, and thus q E R. In both cases (a) and (c) we therefore can use the equality O r (I) = R.  (c)^(a). Now we can take I -1 as K. Using (c), the inclusion O r (I)R C O r (I) can be written in an equivalent form O r (/)/ -1 / C Or (I). By the definition of L it follows that O r (/)/ -1 C L, and thus O r (I) C LI, i.e.,  RC LI. Consequently R= LI, and (a) holds. (a)^(c). This time we can put K = L. Also, from the definitions of O r (I), and L it follows that LI C I C O r (/). Hence R C I C R, i.e., (c) holds. Finally, the remark above shows that either of (a), (b), or (c)  implies R O r (I). ^  Non commutative Priifer rings and sone generalizations ^14 Proposition 2.1.1 The following are equivalent for a ring R: (a) R is a right semi-Priifer (right w-semi-Priifer, or right w-Priifer, or right Priifer) ring; (b) R is a semiprime Goldie (semiprime right Goldie, or prime right Goldie, or prime Goldie) ring, and every lg. fractional right ideal of R is a progenerator of Mod-R; (c) R is a semiprime Goldie (semiprime right Goldie, or prime right Goldie, or prime Goldie) ring, and R has property (P). Proof. We give a proof only for the case where R is a semiprime right Goldie ring. (a) .#>. (b). By Lemma 2.1.1 and Lemma 2.1.2. (b)  (c). Since R is a semiprime right Goldie ring, every essential right  ideal of R contains a regular element of R by Proposition 1.2. Therefore every f.g. essential right ideal of R is a fractional right ideal. (c)^(b). Let I be a f.g. fractional right ideal of R. From the definition of a fractional right ideal, we know that there exist regular elements c and d of Q such that c E I and dl C R. Then dc E dI , and de is a regular element of R. Hence dl is a f.g. essential right ideal of R by Proposition 1.2. Then (c) implies that dI is a progenerator of Mod-R. But we have IR (di )R, so IR is a progenerator of Mod-R. 0 Example 2.1.1 Let F be a field such that there exists an isomorphism A of F onto a proper subfield of F. Let R be the abelian group consisting of all polynomials in x with coefficients from F, with coefficients written on the right. Define a multiplication in R by using the rule ax" = x"(A"a) for all  Non commutative Priifer rings and sone generalizations ^15 a E F and all n. Then the ring R is a principal right ideal domain, and R is right Ore but not left Ore [17, Ex.1, P101]. Hence R is right Goldie but not left Goldie. Therefore we have (a) R is a right w-Pritfer ring; (b) R is not a left w-semi-Priifer ring; (c) R is not a right semi-Priifer ring.  The example also tells us that being a w-Priifer ring (or a w-semi-Priifer ring) is not a left-right symmetric concept. Since it will be shown that a Prfifer ring or semi-Prfifer ring is left-right symmetric and a ring is a semiPriifer ring if and only if it is a finite direct sum of Prfifer rings, we have the following implication diagram: semi-Prfifer  0(7  ^  \s c ^ Priifer right w-semi-Priifer j  right w-Priifer Proposition 2.1.2 Every right w-semi-Prilfer ring is a right and left semi-  hereditary ring.  Proof. Suppose R is a right w-semi-Priifer ring and /R a f.g. right ideal of R. We have a right ideal J of R which is maximal with respect to I fl J = 0.  And I J = I ED J< e RR. Since R is a semiprime right Goldie ring, I 1ED J contains a regular element r of R by Proposition 1.2. Write r = a + b, a E I and b E J, and let K = I ED bJ. Then K is a f.g. essential right ideal of R. By Proposition 2.1.1 KR is projective, and so is IR. We have shown that R is a right semihereditary ring. Because the property of being a semiprime  Non commutative Priifer rings and sone generalizations ^16 right Goldie ring is Morita invariant, M n (R) End(Rn) is a semiprime right Goldie ring for all n, and thus Mn (R) does not possess an infinite set of orthogonal idempotents. Hence R is left semihereditary by Theorem 1.7. ^  Lemma 2.1.3 If a ring R is a right w-semi-Priifer ring, then M n (R) is a right w-semi-Prifer ring for every n. Proof. Since the property of being a semiprime right Goldie ring is Morita invariant, Mn (R) is a semiprime right Goldie ring. It is also clear that Mn (R) is a semihereditary ring because of Proposition 2.1.2 and Theorem 1.7. So it suffices to show that every f.g. essential right ideal L of Mn (R) is a generator of Mod-Mn (R). We need some notation: if A is a subset of R, set A[k] {(a z ,) E Mn (R) : a z , = 0 Vi k; ak3 E Al. It is easy to see that L mn (R) = (CiiL)mr,(R) (e22L)Mn (R)  e • • • e  (en.L)m„(R),  where ekk is the matrix having a lone 1 as its (k, k)-entry and all other entries 0, and for each k ( 1 < k < n ), there exists a right ideal I k of R such that (ekkL)m„(R) = Ik [k]. If I is a nonzero right ideal of R, then /[1] is a nonzero right ideal of Mn (R), so L fl = (I fl  ION  0. This implies  that I fl I l 0. Hence / 1 is a f.g. essential right ideal of R. By Proposition 2.1.1, / 1 is a generator of Mod-R. We know R is Morita equivalent to Mn (R) via the Morita equivalence G = (— — Om n (R)Rn)R : Mod-Mn (R) —4 ModR. In particular, G((elinmn (R)) (eiiL Omn (R) Rn)R. But we have a R-homomorphism cb : (eiiL Omn (R) R n )R^)R which is defined by  Noncommutative PrUfer rings and sone generalizations ^17  / a l . •^an \ 0^0 0^...  X1  aixi + • •^ci n x„.  (8) (  xn  Obviously 0 is onto. Since (h ) R is a generator of Mod-R, we infer that (e n L  mn ( R )  Rn) R is a generator of Mod-R. Hence (e li n mn ( R ) is a generator  of Mod-Mn (R) by [2, Prop.21.6]. Thus we have L is a generator of ModMn (R) because e n ', is an image of L as right Mn (R)-modules. ^ Lemma 2.1.4 Let R be a right w-semi-Priifer ring, e an idempotent of R with ReR = R. Then eRe is a right w-semi-Priifer ring.  Proof. Clearly eRe is a semiprime right Goldie ring. Suppose L is a f.g. essential right ideal of eRe, we want to show that L is a generator of ModeRe. Write L E,(ex i e)eRe. Then L = LeRe = Te, where T LeR is a  f.g. right ideal of R. Clearly T C eR. We claim that TR< e (eR) R . In fact, if 0 er E eR, then erRe 0 since R is a semiprime ring. Hence erRe is a nonzero right ideal of eRe. Thus erRe fl L 0, i.e., 0 erxe E L for some x E R. So 0 erxeR C T. Next we show that TR is a generator of Mod-R.  We know ((1 — e)R)R has finite Goldie dimension, and so there exist nonzero uniform right ideals U, of R such that  u,  + • • • +^e • • e  uri<e((1 — e)R)R.  We claim U,eR 0 Vi. Otherwise eR C^fR for some idempotent f E R by Proposition 2.1.2 and Theorem 1.6. Since fR is an ideal, we have Rf C fR and so (1 — f)Rf = 0. Since R = Rf R(1 — f), it follows  Noncommutative Priifer rings and sone generalizations^18  that R(1 — f) is a two-sided ideal, and hence f R(1 — f) is a right ideal. Now [fR(1 — f)] 2 0, and R has no nonzero nilpotent right ideals, hence f R(1 — f) = 0. Given any r E R, we thus have fr(1 — f) = 0 as well as  (1 — f)rf = 0, whence fr = frf rf. Then R. ReR = RIR = Rf, and this implies that f = 1. Therefore U, = Ui R = 0. The contradiction shows that Ui eR 0 Vi. Thus eRUi 0 Vi since R is a semiprime ring. Since TR < e (eR)R, we have T n eRUi 0. Then 0 (T  n eRUi) 2  C eRUT. So  UiT 0 Vi. For each i, choose an ai E Ui such that ct,T 0. Then ct i T + • • • + a n T a i T ED •  e a n T < e^— e)R)R.  Therefore T ED a i T e•-e a n T <, eRe  (1-^=  By Proposition 2.1.1, T Eh a 1 T • • • a n T is a generator of Mod-R. Since each ct,T is an image of  TR,  we conclude that T is a generator of Mod-  R. To see L is a generator of Mod-eRe, we use the Morita equivalence  HomR(eR, --) : Mod-R Mod-eRe. Since T is a generator of Mod-R, we have L e R e^(Tie)i eRe^(HomR(eR,TR))eRe (by [2, Prop.4.6]) is a generator of Mod-eRe by [2, Prop. 21.6]. Finally, since R is a semihereditary ring, every f.g. submodule of (eR) R is projective. Therefore eRe =Hom R (eR, eR) is a right semihereditary ring by [2, Prop.21.6; Prop.21.8]. ^ Theorem 2.1.1 The property of being a right w-semi-Pritfer (right semiPricier, right w-Pritfer, or right Priifer) ring is a Morita invariant.  Proof. Suppose R is a right w-semi-Priifer ring which is Morita equivalent to ring S. Then S eM7,(R)e for some n and some idempotent e E Mn(R)  Non commutative Priifer rings and sone generalizations ^19  with M,,(R)eM„(R) = 1117,(R) by Theorem 1.5. By Lemma 2.1.3 and Lemma 2.1.4, S is a right w-semi-Priifer ring. Since the properties semiprime Goldie, prime right Goldie, and prime Goldie are all Morita invariants, the other parts follow immediately.  ^  Proposition 2.1.3 The ring R is a right semi-Prizfer (right w-semi-Prifer,  or right w-Prizfer, or right Pricier) ring if and only if R is a semiprime Goldie (semiprime right Goldie, or prime right Goldie, or prime Goldie) ring and every f.g. essential submodule of each progenerator of Mod-R is a progenerator of Mod-R.  Proof. One direction is clear by Proposition 2.1.1. Suppose that R is a right w-semi-Priifer ring. Let PR be a progenerator and NR a f.g. essential submodule of PR. And set S End(PR). Then we have the Morita equivalence F = Hom R ( s PR , --) : Mod-R Mod-S. By [2, Prop.21.6; Prop.21.8], F(N)s is a f.g. essential submodule of F(P)s = Ss. We know S is a right w-semi-Priifer ring from Theorem 2.1.1. Hence it follows that F (N) s is a progenerator of Mod-S from Proposition 2.1.1. Therefore NR is  a progenerator of Mod-R by [2, Prop.21.6; Prop.21.8].  ^  Proposition 2.1.4 The following are equivalent for a ring R:  (a) R is a right w-Pritfer (or right Priifer) ring; (b) R is a right Goldie (or Goldie) ring and every f.g. nonzero right ideal of R is a progenerator of Mod-R; (c) R is a right Goldie (or Goldie) ring and every f.g. nonzero submodule of each progenerator of Mod-R is a progenerator of Mod-R.  Noncommutative Priifer rings and sone generalizations^20 Proof. (a)^(b). By Proposition 2.1.2, it is enough to show that every f.g. nonzero right ideal I of R is a generator. We can find a right ideal J of R such that^RR. Since R is a right Goldie ring, there exist uniform submodules J 1 ,^, Jt of JR such that J1 ED • • • e^<, JR. Hence  Jt <, RR. Since R is prime, JtI^0 for each i. So we can choose some ai E Ji with ail 0. Then I e a l l e • • - W a t I < e RR. By /^+ • • • e  Proposition 2.1.1, / ED al/ ED • • • ED a t I is a generator of Mod-R. Therefore I is a generator of Mod-R.  (b)^(a). That every f.g. nonzero ideal of R is a generator implies that R is a prime ring. (b)  (c). Similar to the proof of Proposition 2.1.3.  (c)  (a). By Proposition 2.1.1. ^  Proposition 2.1.5 The ring R is a right w-semi-Pritfer ring if and only if Z(RR ) = 0, RR is finite-dimensional and R has Property (P). Proof. One direction is clear. Suppose that Z(RR) = 0, RR is finitedimensional, and R has Property (P). We only need to show that R is semiprime right Goldie. Suppose / 2 0 for an ideal I of R. We have a right ideal J of R such that lef  <, RR. Then (/+J)/  C JI C I f1 J  = 0.  Since RR is finite-dimensional, there exist f.g. right ideals I 1 ,J1 of R such that <, IR,  Jl  <e JR. Therefore we have / 1 (1) J1 <, RR. Since R has Property  e J1 is a generator of Mod-R. Thus RR is an R-homomorphic image of^e J1 for some n. Noting that (II e Jo/ 0, we have I = RI = 0. (P), I l  ) ()  Therefore R is semiprime. By Theorem 1.2, R is a right Goldie ring. ^  Noncommutative Prffer rings and some generalizations ^21 Proposition 2.1.6 The ring R is a right w-Prüfer ring if and only if RR is  finite-dimensional and every f.g. nonzero right ideal of R is a progenerator of Mod-R.  Proof. The necessity follows from Proposition 2.1.4. For the converse, it is easy to see that R is a prime ring. Suppose Z(RR) 0. We can choose a f.g. right ideal I of R such that I C Z(RR ). By our assumption, I is a generator of Mod-R, and thus RR is an epimorphic image of /(n) for some n by Proposition 1.3. Since /(n) is singular, we have that RR is singular. This is a contradiction since 1 Z(RR). Therefore Z(RR) = 0, and thus R is a prime right Goldie ring by Theorem 1.2. ^ Some other characterizations of right w-semi-Priifer (right semi-Priifer, right w-Priifer, or right Prfifer) rings will be presented in the next chapter.  2.2 A structure theorem and further properties of right w-semi-Priifer rings Lemma 2.2.1 Let R be a right w-semi-Prifer ring, and Q Qrd (R). If e  is a central idempotent of Q, then (eR)R is a projective R-module.  Proof. Write e^1 — e = u 2 v -1 , where u, E R and v E CR(0). Define a map 0 : eR ED (1 — e)R^R by 0(ex + (1 — e)y)^u t x u 2 y Vx, y E R. Suppose ex + (1 — e)y ex' + (1 — e)y'. Then ex = ex', i.e., (u i v -1 )x = (u i v')x i . So u i x = v[(u i v -1 )x] = v[(u i v -1 )x'] = u i x i . Similarly u 2 y = u 2 y'. Hence u i x u 2 y = u2y'. Thus .0 is well defined. Clearly 0 is a right R-module homomorphism. If u i x u 2 y = 0, then 0 =  Noncommutative Priifer rings and some generalizations^22 (u i v -1 )vx (u 2 v -1 )vy v[(u i v -1 )x (u 2 v -1 )y], and then ex + (1 - e)y 0. So 0 is one to one. Therefore we have (eR + (1 - e)R) R Im4). But /m0 is a f.g. right ideal of R, and so it is projective by Proposition 2.1.2. Hence (eR) R is projective. ^ Proposition 2.2.1 Let R, Q be as above, e any central idempotent of Q. Then e E R. Proof. Since (1 - e)R is a right projective R-module, the exact sequence 0 -+ eRn R^(1 - e)R 0 splits. Then eR n R is a direct summand of RR. So we have eRnR= fR for an idempotent f E R. Then fQ C eQ. If 0 e E eQ, write^ac-1 for some a E R and c E CR(0). Then 0 (e)c = ea E eR. Write ea = uv -1 for some u E R and v E CR(0). We have 0 (e)cv eav u E eR  nR  fR, and so 0 (e4- )cv E fQ. Therefore  (fQ) Q <, (eQ) Q . Since Q is a semi-simple Artinian ring, (fQ) Q is a direct summand of (eQ) Q . It must be that fQ = eQ. Then e=fe=ef^since e is central. ^ Proposition 2.2.2 Let R be a right w-semi-Pritfer ring, Q^Qrd (R) ,(21 e • • + Qn, where each Q i is a simple Artinian ring. Then R = (R n  WEB- • -ED (RnQ,,), each RnQi is a right w-Priifer ring and Qrd (RnQi) = Q i . Proof. By Theorem 1.1, Q cri (R) is a semi-simple Artinian ring. Hence the Wedderburn-Artin theorem asserts that Q rc i(R) is a finite direct sum of simple Artinian rings: Qrci (R)^Q i El) • • • ED (4, with each Q, being a simple Artinian ring. We have 1 R^1Qrc1 (R) = 1Q, + • • • + 1 Q „, where 1 Q , is the identity of Q. Set R, = Rn Q,. Then 1 Q E R, by Proposition 2.2.1. Hence Ri is a  Non commutative Priifer rings and some generalizations^23 subring of  Q.  It is straightforward to check that each Ri is a right order of  Q. So R, is a prime right Goldie ring. Each Ri is obviously an ideal of R, and for every x E R, x xlcji + • • • + x1Q„ E Rl  e • • • ED  fin . Therefore we  have R R 1 ED • • • ED R n . To see each Ri is a right w-Priifer ring, we only need to show that R, has property (P) by Proposition 2.1.1. Let Li be a f.g. essential right ideal of R, and let I = R 1 + • • + Ri-i • +Rri R 1 ED • ED L ED • • • ED R. Then I is a f.g. essential right ideal of R, and so II? is a progenerator of Mod-R by Proposition 2.1.1. Hence (I i ) R is projective, and this implies that (/,)R, is projective. On the other hand, if f E Hom(IR, RR), we have f(R3 ) C R3 , if j i, and f(I,) C Ri . Since  I R  is a generator of  Mod-R, R --= feHom(IR ,R R ) I m f =  (E f EHorn(IR ,R R )  f (Ri)) ED • • •  (Efewom(IR,RR) f(h)) ED • • • ED (EfeHovi(IR ,R R )  f  (Rn))•  So we have >fEHom(IR,RR)  1 (hi) =  EfEHorn(LR,R,R) f(L)^EfEHon-t(I, R ,,R, R ,)  f i)  It follows that /, is a generator of Mod-Ri. Hence / i is a progenerator of Mod-R1. We can conclude that each Ri is a right w-Priifer ring. ^ Theorem 2.2.1 A ring R is a right w-semi-Priifer ring if and only if it is  a finite direct sum of right w-Priiler rings. Proof. The necessity follows from Proposition 2.2.2. Suppose R E  [r_ i R be a direct sum of right w-Priifer rings Ri. Then i  Crd (R)^Qrd(R,) which is a semi-simple Artinian ring. Hence R is a  Noncommutative PrUfer rings and some generalizations^24  semiprime right Goldie ring. Suppose I is a f.g. essential right ideal of R. Let 7i be the i ll' projection of R onto Ri. We have 0 I fl R, C 7r 1 (IR ), and this implies that 70) is a f.g. essential right ideal of R,. By Proposition 2.1.1, ir i (/) generates R, as a right R 2 -module, and thus 7ri(/) generates R, as a right R-module. Therefore we have shown that IR is a generator of Mod-R. Next instead of proving IR is projective, we show R is a right semihereditary ring. For each m, let Mm (R) = Mm (R i ) e • • • e mn,(R7i). Given x E Mm (R), write x = x 1 + • • • + x„, with each x, E Mm (Ri ). We want to show that xM„,(R) is a projective right Mm (R)-module. Since each Mm (Ri) is still  a right w-Priifer ring, we can assume m = 1. Since xiRi is a projective right R 2 -module, we have (ROI?, (x,Ri)R,  e Ui for some right R i -module  U1 . We know U1 can be regarded as a right R-module canonically. Thus as right R-modules we still have (R1)R ti(xiRi)R  ® U,. Therefore (x,Ri)R is  projective since (R 1 ) R is. Then xR = x 1 R • xn,R  e • • • e  x.Rn  is a projective right R-module. We have actually shown that xM,,(R) is a projective right Mm (R)-module for every x E Mm (R). By Proposition 1.5, R is a right semihereditary ring. ^ Proposition 2.2.3 If R is a right w-Priifer ring, e a nonzero idempotent, then eRe is a right w-Prifer ring.  Proof. Since eR is a progenerator of Mod-R by Proposition 2.1.4, then eRe End(eR) is Morita equivalent to R, thus is a right w-Priifer ring by Theorem 2.1.1. ^ Corollary 2.2.1 If R is a right w-semi-Pritfer ring, e a nonzero idempotent, then eRe is a right w-semi-Pritfer ring.  Noncommutative Prlifer rings and some generalizations^25  Proof. By Theorem 2.2.1 and Proposition 2.2.3. ^ By a complete set of idempotents of a ring we mean a set of pairwise orthogonal idempotents: {e l , • • • , e t } with EL I e i = 1. Proposition 2.2.4 If R is a right w-Prifer ring, then there exists a com-  plete set of idempotents e 1 ,- • •,e,,, such that R = e i R ED • • ED e n,R and for each e i Re i is a right w-Prifer domain which is Morita equivalent to R.  Proof. Since R is a prime right Goldie ring, RR has the ascending chain condition (and the desending chain condition) on the set of direct summands of RR (see [2, Ex.§10.11]). By [2, Prop.10.14; Prop.7.2], there exists a complete set e 1 , • • • , e m of idempotents in R such that R = ei R ED • • • ED e m R and each e i R is indecomposable as a right R-module. We know e i R is a progenerator by Proposition 2.1.4. Therefore R is Morita equivalent to e i Re i End R (e,R). By Proposition 2.2.3, e i Re i is also a right w-Priifer ring. Now let 0 x E eiRei, then xeiRe i is a projective e i Rei-module and it follows that x 1 fe,Rei for some f 2 = f E e i Rei. But the ring e i Re i has exactly one nonzero idempotent, namely e i . It follows that f 0 or e,. Since x 0, it follows f = 0, i.e., x 1 = 0 for all 0 x E eiRei. Therefore e i Re i is a domain. ^ Lemma 2.2.2 Let M1 ED M2 ED • • • ED Mn = A ED B be a decomposition in  Mod-R such that End(A R ) is a local ring. Then there exists i,1 < i < n, and an isomorphism Mi A ED X for some X E Mod-R.  Proof. See [9, P39-40]. ^  Noncommutative Priifer rings and some generalizations^26 A module MR is called a quasi-injective module if for each submodule  N of M, every R-homomorphism from N into M can be extended to an R-homomorphism from M into M.  Proposition 2.2.5 Let R be a right w-Priifer ring. If there exists a nonzero f.g. quasi-injective projective right R-module, then R is a simple Artinian ring. Proof. As in Proposition 2.2.4, R = e 1 R  e • • • e  e r,R, where each e i R is  an indecomposable R-module. Let MR be a f.g. quasi-injective projective module. Since e l R is a generator, there exist an integer m > 0 and some R-module X such that (e i R)"1 M  e X. Since R is finite-dimensional, e 1 R,  hence (e i R)m has finite Goldie dimension. So M has finite Goldie dimension. Write M = M1 ED• • ED Mk, where each M, is an indecomposable submodule of  M. Now, if M is a quasi-injective module, then each Mi is a quasi-injective module. Therefore End(M 1 ) is a local ring. Thus Lemma 2.2.2 implies that e 1 R tiM 1 U, for some U. As we know e 1 R is an indecomposable Rmodule, we have e 1 R 2,-1 MI, is quasi-injective. We can also show that each  e,R M1 . Therefore RR M in is a quasi-injective module. Now Baer's Criterion implies that R is a right self-injective ring. Then R = E(RR) is a semi-simple ring by [18, Th.4.28]. Hence R is a simple Artinian ring. ^ We know that Z, the ring of integers, is a Priifer ring, but not a simple Artinian ring. We also know that Q, the field of rational numbers, cannot be embedded in Z(') for any index set I. The following is one way to see this:  Corollary 2.2.2 If R is a right w-Pritler ring, but not simple Artinian, then, for any f.g. right R-module M, the injective hull E(M) of M cannot  Noncommutative PrUfer rings and some generalizations^27 be embedded in a free R-module. Proof. Suppose M is a f.g. R-module, and 0^E(M)^R(1) is exact for some I. Since MR is finitely generated, l(M) C R (F) , where F is a finite subset of I. Let p : R( 1) R(F) be the canonical projection. We consider Ker(p o 1). Since Ker(p o 1)  nM=  0, and M <, E(M), we conclude that  Ker(p o 1) = 0. Thus E(M) is embedded in R( F ). But E(M) is injective, so it is a direct summand of R( F ), and therefore finitely generated. Now the previous proposition implies that R is a simple Artinian ring. ^  2.3 Priifer rings and semi-Priifer rings Proposition 2.3.1 Let R be a right semi-Priifer ring, and Q',. i (R)^Qi Eh • • •  e  Q„, where each Q i is a simple Artinian ring. Then R  (Rncme • • •e  (R n Q,„), where each R n Q, is a right Priifer ring and Q7 1 (R n Qi) = Qi. Proof. Similar to the proof of Proposition 2.2.2. ^ Theorem 2.3.1 A ring R is a right semi-Priifer ring if and only if R is a finite direct sum of right Priifer rings. Proof. Similar to the proof of Theorem 2.2.1. ^ Next we turn to the left-right symmetry of Priifer rings and semi-Priifer rings. Theorem 2.3.2 [1, Prop. 1.14 A ring R is a right Priifer ring if and only if R is a left Priifer ring.  Noncommutative Prlifer rings and some generalizations ^28  Proof. Suppose R is a right Priifer ring. We want to show R is left Priifer. We know R is a Goldie and left semihereditary ring by Proposition 2.1.2. So, to show R is a left Priifer ring, it suffices to show that for any f.g. nonzero left ideal J of R, J is a generator of R-Mod by using the left version of Proposition 2.1.4. Since R is left semihereditary, J is a projective left R-module. So we may assume that Rn J N for some n and some N E R-Mod. Therefore we have R J RTh f for some idempotent f E EndR(Rn) = Mn (R). Since Rn is a progenerator of R-Mod and Mn (R) = End R (Rn), we have a Morita  equivalence Rn ®m„(R) : Mn (R) —Mod R-Mod. As left R-modules,  R n O mn ( R ) Mr,(R)f (Rn f) J (via a b ab). So R J is a generator of R-Mod if and only if mn (R)(M7,(R)f) is a generator of M n (R)-Mod by [2, Prop.21.6]. Also, we know that M n (R) is a right Priifer ring from Theorem 2.1.1. Therefore, without lose of generality, we may assume that J = Re for some idempotent e of R. Since R is a prime ring, ReR <, RR. Then ReR  n cR (o) is not empty by Proposition 1.2. Thus there exist elements  r i , t, E R (i = 1, • • • , m) such that x = r i et i + • • • + r m et n, E CR (0). Consider right ideals I = r i eRd-• • .+7•„„eR and P = f Rd-x R. Then I is a f.g. fractional right ideal of R, and so 1'1 = R, since R is right Priifer. On the other hand, P C ReR, and I = r i eR • rm eR C r i eReR + • • + r ni eReR = I eR C TP. From P C ReR and I C IP, it follows that R C I-11p C I -1 I ReR = RReR C ReR. Hence R = ReR. Because the trace ideal T( R Re)  is a two-sided ideal of R and T(RRe) E{O(Re) : q E Hom( R Re, R R)} D Re, we have R ReR C T(RRe) C R. Therefore R = T(RRe). By (the left version of) Proposition 1.3, J Re is a generator of R-Mod.  ^  Noncommutative Priifer rings and some generalizations^29 Corollary 2.3.1 The ring R is a right semi-Pritfer ring if and only if R is  left semi-Pritler.  Proof. By Theorems 2.3.1, 2.3.2. ^ From now on we will use the terms Priifer ring and semi-Priifer ring instead of right (or left) Priifer ring and right (or left) semi-Priifer ring respectively. Proposition 2.3.2 If R is a Priifer ring and e is a nonzero idempotent,  then eRe is a Priifer ring.  Proof. By Proposition 2.1.4, eR is a progenerator of Mod-R, and so eRe End(eR R ) is Morita equivalent to R. It follows that eRe is a Priifer ring from Theorem 2.1.1. ^ Corollary 2.3.2 If R is a semi-Prilfer ring and e is a nonzero idempotent,  then eRe is a semi-Pritfer ring.  Proof. We may assume that R^e  R2 ®• • •  Rn with each Rt a Priifer  ring and e = e i^• • + et with t < n and each e t a nonzero idempotent of Then eRe^e 1 R 1 e 1 e • • • e e t R t e t . The previous proposition implies that each e,R t e t is a Priifer ring. Hence Theorem 2.3.1 implies that eRe is a semi-Priifer ring. ^ Proposition 2.3.3 [1]. Each overring of a Pritfer ring is a Pritfer ring.  Proof. Let R be a Priifer ring, Q^Q ct (R), and let S be an overring of R, i.e., S is a subring of Q such that R C S C Q. Clearly S is right and left order in Q. Hence S is a prime Goldie ring with Q d (S) = Q. Suppose that J  Non commutative Priifer rings and some generalizations^30  is a f.g. fractional right ideal of S, e.g., J a l S-D• •-kar,S. We may assume that a 1 is a unit of Q. Consider I = a i Rd- • • • + a ri l?. Then I is a fractional right ideal of R, and thus / -1 / R and // -1 = 0/(/). Now we have IS = J and S = RS = .1-"J. The last equality implies that T(Js) = S, i.e., J is a generator of Mod-S. Therefore, by Lemma 2.1.2, J -1 J = S. On the other hand, JI -1 J = JRS = J. This implies that / -1 C J. Hence 1 E^--=^C JJ -1 . Then 01(J) C (01(J)J)J -1  C JJ -1 .  Therefore JJ -1 01(J). It follows that S is a Priifer ring. ^ Corollary 2.3.3 Each overring of a semi-Prifer ring is a semi-Priifer ring.  Proof. Let R be a semi-Priifer ring, and Q d (R) = Q i  ED • • •  ED Q,,, where each  Q i is a simple Artinian ring. Then by Proposition 2.3.1, R (Rn Q 1 ) ED • • • e  (R n (2,), and each Rn Q i is a Prfifer ring with Qrd (Rn Q,) = Q i . Now if S is an overring of R, then S n Q i is an overring of R n Q. By Proposition 2.3.3, S  n Qi is a Priifer ring. But it is easy to see S  (S  n Q i ) ED • • • ®(S n  Qn)•  It follows from Theorem 2.3.1 that S is a semi-Prfifer ring. ^ The concept of a prime Dedekind ring was first introduced by Robson in [32] by the term "maximal order". An important characterization of the prime Dedekind rings of Robson is stated as follows: A ring R is a prime Dedekind ring if and only if every nonzero submodule of a (left or right) progenerator is also a progenerator [29, Th.2.10, P.140]. It was proved in [1] that a ring is a Priifer ring and a bounded Krull ring if and only if it is a prime Dedekind ring, where a bounded Krull ring is defined in the sense of Marubayashi (cf.[27] Sec.1). Theorem 2.3.3 The following are equivalent for a ring R:  Non commutative Priifer rings and some generalizations^31 (a) R is a Priifer and (both sides) Noetherian ring; (b) R is a prime Dedekind ring.  Proof. (b)^(a). Every one-sided ideal of R is a progenerator, and hence a f.g. R-module. It follows that R is a Noetherian ring. Thus Proposition 2.1.4 implies that R is a Priifer ring. (a)^(b). Let PR be a progenerator and We want to show that  NR  NR  nonzero submodule of  PR.  is also a progenerator. Let S = EndPR. Then R  is Morita equivalent to S via the Morita equivalence F = Hom R ( s PR ,--): Mod-R -->. Mod-S, and F(N)s is a nonzero right ideal of S. S is also a Priifer ring. Since the property of being a one-sided Noetherian ring is a Morita invariant, S is a right Noetherian ring. So F(N) s is a nonzero f.g. right ideal of S. By Proposition 2.1.4, F(N) s is a progenerator of Mod-S. Hence Ns is a progenerator of Mod-R. So R is a prime Dedekind ring. ^ Corollary 2.3.4 A ring is a semi-Prifer Noetherian ring if and only if it is a finite direct sum of prime Dedekind rings.  Proof. This follows from Theorems 2.3.1, 2.3.3. ^ A ring R is right bounded if every essential right ideal of R contains an ideal which is essential as a right ideal. Note that a prime ring R is right bounded if and only if every essential right ideal of R contains a nonzero ideal. A right and left bounded ring is called a bounded ring. Proposition 2.3.4 The ring R is a right bounded semi-Priifer ring if and only if R is a finite direct sum of right bounded Priifer rings.  Noncommutative Priifer rings and some generalizations ^32  Proof.^By Theorem 2.3.1, R^  e • • • e  Rii , with each R, being a  Priifer ring. Given an essential right ideal /i of Ri. Then I^EP • •  •  R1_1 ®Ii e Ri + i ® • • ED R -, is an essential right ideal of R. Hence there exists 7  an ideal J of R such that J  C  I and JR < e RR. J n R, is an ideal of R i , and  we have 0 JnR i cInR i cL. Therefore Ri is right bounded. (-). Let R = R1  E•••®  R„, where each R i is a right bounded Priifer  ring. Then R is a semi-Prfifer ring by Theorem 2.3.1. Suppose /R is an essential right ideal of R. We need to show that IR contains an ideal of R which is essential as a right ideal. It is easy to see that I n Ri <e (Ri)R,• Hence for each i there exists a nonzero ideal K, of Ri such that Ki cIn R, and (KO)R, (Ri)R t . Hence (Ki)R (ROR• Set K K1 ED • • • EDKn . Then K is an ideal of R, K C I and KR <E RR.  ^  A module MR is faithful if for every 0 r  E R, Mr  0. Every generator  is faithful. But the converse is not true. We call a ring a right FPF ring if every f.g. right faithful module is a generator. An FPF ring is defined to be a left and right FPF ring. There are some known relations between bounded prime Dedekind rings and prime FPF rings. In fact a bounded prime Dedekind ring can be characterized as a Noetherian prime right (or left) FPF ring [10, Th.4.6]. In the following we point out how a bounded Priifer (or semi-Priifer) ring is related to an FPF ring. Theorem 2.3.4 For a ring R, the following are equivalent: (a) R is a prime right FPF right semihereditary ring; (b) R is a right bounded Priifer ring; (c) R is a prime right FPF left semihereditary ring.  Non commutative Priifer rings and some generalizations^33  Proof. (b)^(a)&(c). If R is a right bounded Priifer ring , then R is prime Goldie semihereditary ring. Moreover, every f.g. nonzero right ideal is a generator by Proposition 2.1.4. Now, by [10, Th.4.7], R is a prime right FPF ring. So (b) implies (a) and (c).  (a)^(b). If R is a prime FPF right semihereditary ring, then, by [10, Th.4.7], R is a right bounded Goldie (both sides) ring and every nonzero f.g. right ideal is a generator. Now, since R is also a right semihereditary ring, it follows that every nonzero f.g. right ideal of R is a progenerator. By Proposition 2.1.4, R is a Priifer ring. (a)^(c). From the proof above, we know any ring R which possesses (c) must be a right bounded prime Goldie ring for which every f.g. nonzero  right ideal is a generator. Since the property of being a prime Goldie ring is Morita invariant, Mn (R) is prime Goldie ring for all n. In particular, Mn (R) does not possess an infinite set of orthogonal idempotents. So Theorem 1.7 implies that R is right semihereditary ring, and so (a) holds. ^ Corollary 2.3.5 For a ring R, the following are equivalent:  (a) R is a prime FPF left semihereditary ring; (b) R is a bounded Priifer ring; (c) R is a prime FPF right semihereditary ring. ^ Corollary 2.3.6 Every right bounded semi-Prilfer ring R is a semiprime  semihereditary right FPF ring; The converse is true if R also has ACC on annihilators.  Proof. For the first part, it is enough to show that R is a right FPF ring. By Proposition 2.3.4, R = R l ED • • • ED R„, with each R, being a right bounded  Noncommutative Prffer rings and some generalizations^34 Prfifer ring. Then each Ri is a right FPF ring by Theorem 2.3.4. Therefore we have R is a right FPF ring by [10, Th.3.4]. For the second part, we first note that R is a Goldie ring by [10, Cor.3.16C]. Then [10, Th.3.4(1)] implies that R R 1 W • • •W R,,, where each R i is a prime right FPF ring. Since R is a semihereditary ring, it is easy to show that each R i is also a semihereditary ring. By Theorem 2.3.4, R i is a right bounded Priifer ring. Now Proposition 2.3.4 implies that R is a right bounded semi-Prfifer ring. ^ It was proved that every Prfifer ring is Morita equivalent to a Priifer domain in [1, Th.2.3]. We give the following stronger result: Theorem 2.3.5 Let R be a Pricier ring. Then there exists a complete set of idempotents el, • • • , e n such that R eiR ED e2R ED • • • ® e r,R, where for each e i R is a uniform R-module, e i Re i is a Prifer domain and R is Morita equivalent to e i Re i . Proof. By Proposition 2.2.4, it is enough to show that eR is a uniform right R-module for each indecomposable module eR. Let N be a nonzero R-submodule of eR. We want to show that N is an essential submodule of eR. Suppose N fl K 0 for a submodule K of eR. We know that there exists a submodule L of eR which is maximal with respect to K C L and N fl L 0. By Proposition 1.1, N is embedded in eR/L as an essential submodule. From Theorem 1.2, we have Z(RR) = 0, and thus Z(NR) 0. It follows that Z(eR/L) = 0. By noting Proposition 1.2, we have that eRI L is a f.g. torsionfree right R-module. Therefore, it follows from Theorem 1.3 that eR/L is embedded in a f.g. free right R-module. Since R is Prfifer, eR/L is a projective right R-module. Thus eR L ED (eR/L). Now the  Non commutative Prater rings and some generalizations ^35  indecomposablity of eR implies that L 0. Hence K 0, and N is essential in eR. ^ Lemma 2.3.1 Let R = R 1 e• • •e^and S = Sl  e• •eSn.  If R i is Morita  equivalent to S i (i^1, • • , n), then R is Morita equivalent to S.  Proof. Well-known. ^ Corollary 2.3.7 Every semi-Priifer ring is Morita equivalent to a finite direct sum of Priifer domains.  Proof. By Theorem 2.3.1 , Lemma 2.3.1 and Theorem 2.3.5. ^ In the final part of this section, we consider semiperfect Priifer rings. A ring R is semiperfect if RI Rad(R) is semi-simple, and idempotents of RI Rad(R) lift. By a theorem of Bass, the ring R is semiperfect if and only  if there exists a complete set of primitive idempotents e l , • • • , e n such that R = e1 R  6) • • • ED e ri R  and each e,Re i is local, where a primitive idempotent  is any idempotent which cannot be written as the sum of two nontrivial orthogonal idempotents (see [2,Th.27.6]). For any semiperfect ring R, there exists a basic set of orthogonal primitive idempotents {e l , • • • , e t } in the sense that for every primitive idempotent f we have R f Re, for exactly one e„ 1 < i < t. In this case e EL 1 e, is called a basic idempotent and eRe is  called the basic ring of R. A module is uniserial if its submodules are linearly ordered with respect to inclusion. A ring R is right serial if RR is a direct sum of uniserial modules. The ring is serial if it is both left and right serial. A local serial ring is called a valuation ring.  Non commutative Priifer rings and some generalizations^36 Lemma 2.3.2 Let R be a Priifer ring, e 2 = e E R. If eRe is a local ring,  then R is a semiperfect ring, and every indecomposable projective right Rmodule is isomorphic to eR, eRe is the basic ring of R, and R L2 Mn (eRe), -  where n is the Goldie dimension of RR (or RR). Proof. By Theorem 2.3.5, R = eiR ED • • • 69 e n R, where n is the Goldie dimension of RR, and e i R is indecomposable for all i. For each i, e i R is a generator and eR is projective module, so we have (e i R) 771 eR  e X for  some m > 0 and some R-module X. Because eRe is a local ring, Lemma 2.2.2 implies eiR^eR Y for some Y. Hence e i R^eR since eiR is indecomposable. And so e i Re i End(e i R)^End(eR)^eRe is local ring.  =  Hence R is a semiperfect ring, and RR 2- (eR)n . So R^End((eR)n)  Mn (eRe). The other assertions follow from [2, Prop.27.10]. ^ Lemma 2.3.3 Let R be a Prifer ring,  e  2  =  e E R. The following are  equivalent: (a) (eR) R is a uniserial module; (b) eRe is a local ring; (c) R(Re) is a uniserial module. Proof. (a)^(b). Let J = Rad(R). We know Rad(eR) = eJ is the intersection of all the maximal submodules of eR. Hence eJ is the unique maximal submodule of eR. By [2, Cor.17.20], eRe is a local ring. (a) (b). If eRe is a local ring, then R is a semiperfect ring by Lemma 2.3.2. By noting a result of Warfield which says a semiperfect semiprime Goldie ring is left semihereditary if and only if it is right serial [38, Cor.4.1,  Noncommutative Prlifer rings and some generalizations^37  we have that R is a right serial ring. Lemma 2.3.2 implies that every uniserial summand of RR is isomorphic to eR. So eR is a uniserial module. (b)^(c). Similarly. ^  Corollary 2.3.8 Let R be a Priifer ring, then R is a left (or right) serial ring if and only if R is a semiperfect ring.  Proof. By Lemma 2.3.3. ^ Theorem 2.3.6 Let R be a Przifer ring. (a) If R is a local ring, then R is a valuation Priifer domain with both RR and RR uniserial modules. (b) If R is a semiperfect ring, then R is a serial ring and R ^M,i (B), where n is the Goldie dimension of R and B, its basic ring, is a valuation Priifer domain.  Proof. (a). By Lemma 2.3.3. (b). This follows from Lemmas 2.3.2, 2.3.3 and Corollary 2.3.8. ^  Corollary 2.3.9 The ring R is a semiperfect semi-Pricfer ring if and only if it is a finite direct sum of matrix rings over valuation Priifer domains.  Proof. (=). By Theorem 2.3.1, R^ED^R„, each R, is a Priifer ring. If R is a semiperfect ring, then every Ri is semiperfect by [2, Coro.27.9]. Therefore we have R, M,,, ; (B1 ) for some valuation Priifer domain Bi by Theorem 2.3.6. (z). A finite direct sum of matrix rings over valuation Priifer domains is clearly a semiperfect ring, and is also a semi-Priifer ring by Theorem 2.3.1. 0  Noncommutative PrUfer rings and some generalizations ^38  Finally, we give a characterization of f.g. torsionfree modules over a semi-Priifer ring. A module MR is flat if whenever f : RN1^RN2 is a monomorphism, we have 1 0 f : M  OR  N1^M OR N2 is a monomorphism.  Proposition 2.3.5 The following are equivalent for a module MR over a semi-Prilfer ring R: (a) M is f.g. torsionfree; (b) M is f.g. flat; (c) M is f.g. projective; (d) M is projective with finite Goldie dimension; (e) M is a finite direct sum of f.g. uniform right ideals of R.  Proof. (a)^(e). By Theorem 1.3, M is a submodule of a f.g. free module FR. Because of Theorem 2.3.1, we may assume R R 1^® R n , where  each Ri is a Priifer ring. By Theorem 2.3.5, each R i is a finite direct sum of f.g. uniform right ideals of Ri. Since every f.g. uniform right ideal of Ri is clearly a f.g. uniform right ideal of R, we have F = E i71, ED/2, where each 4 is a f.g. uniform right ideal of R. Since R is a semihereditary ring, every f.g. R-submodule of 4 is projective. By [24, Prop.8, P85], M Er_ i Ni , with each Ni C (d)^(e). This is because R is a semihereditary Goldie ring. (c) (d). Sandomierski showed in [32, Th.2.1] that if R is a ring such  that Z(RR ) = 0 and PR is a projective module containing a f.g. essential submodule, then P is finitely generated [33, Th.2.1]. Our claim follows. (b)^(c). Well-known.  Non commutative Priifer rings and some generalizations^39 (a)^(b). Suppose MR is flat and let s E CR(0). Define o-, : R —+ R by 0 3(a) = as, which is a monomorphism as left R-modules. This gives rise to -  a commutative diagram  M OR R 1231-- M OR R M --->.^M  where f(x) = xs. Since 1 ® a s is a monomorphism, so is f, and thus x 0 implies xs 0. Therefore M is torsionfree. ^  40  3 Strongly compressible modules Semiprime right Goldie rings constitute a much studied and well known family of rings, and satisfy one of basic conditions satisfied by right w-semi-Priifer (right w-Priifer, semi-Priifer, or Priifer) rings which were defined in chapter 2. Recently LOpez-Permouth, Rizvi and Yousif [26] provided some interesting characterizations of semiprime Goldie rings in terms of their right ideals and of their nonsingular right modules. It was shown that a ring R is semiprime Goldie if and only if every right ideal of R is weakly-injective if and only if  R  is right nonsingular and every nonsingular right R-module is weakly-injective [26, Th.3.9]. This motivates us to look for module-theoretic characterizations of semiprime right Goldie rings. Once such characterizations are established, it can be expected that one can present some new characterizations of right w-semi-Priifer (right w-Priifer, semi-Priifer, or Priifer) rings. In this chapter, we give the definition of strongly compressible modules. It turns out that the concept of strongly compressible modules is closely related to that of weaklyinjective modules and is precisely what we want for our purposes. In fact the connection between strongly compressible modules and weakly-injective modules is similar to that between compressible modules and tight modules (Proposition 3.2.1). We show that a ring  R is semiprime right Goldie if and  only if RR is strongly compressible if and only if every right ideal of  R is  strongly compressible if and only if every submodule of each progenerator of Mod-R is strongly compressible (Theorem 3.1.1). As a corollary of this result, it is shown that a ring  R is semiprime Goldie if and only if every f.g. submod-  ule of the injective hull of RR is strongly compressible if and only if  R is right  Strongly compressible modules^  41  nonsingular and every f.g. nonsingular right R-module is strongly compressible. This characterization theorem can easily imply the above-mentioned characterization theorem of LOpez-Permouth, Rizvi and Yousif because of the strong connection between strongly compressible modules and weaklyinjective modules. In the latter part of the chapter, we apply our results to obtain some new module-theoretic characterizations of prime Goldie (prime right Goldie) rings, and right w-semi-Priifer (right w-Priifer, semi-Priifer, or Priifer) rings, respectively.  3.1 New characterizations of semiprime right Goldie rings Following Jain and LOpez-Permouth [20], a module M is weakly-injective if and only if for every f.g. submodule N of E(M) there exists X C E(M) such that N C X --^±' M. In [23] a module M is said to be compressible if it is embeddable in each of its essential submodules. Definition 3.1.1 A module MR is said to be strongly compressible if for  every essential submodule N of M there exists X C E(M) such that M C X N Every essential submodule of a strongly compressible module is strongly compressible. Every strongly compressible module is clearly compressible. After Theorem 3.1.5, we will give an example of a compressible module which is not strongly compressible.  Strongly compressible modules^  42  Lemma 3.1.1 Every fig. strongly compressible right module has finite Goldie dimension. Proof. Let MR be a f.g. strongly compressible module. Suppose MR is not finite-dimensional. Then there exists an essential submodule N of M such that N EV,?fi Ar” where each Ni 0. Since MR is strongly compressible, there exists a submodule X of E(MR) such that M C N. Then  X = Xi and (Xi)R ====i (NOR for all i. Clearly M C xi for some k. Thus M n  x,  0 for all i > k, contradicting the essentiality of M in E(MR ).  0  Lemma 3.1.2 Let PR be a progenerator of Mod-R. If PR is strongly compressible, then R is semiprime. Proof. Since PR is a progenerator of Mod-R, we can assume that Pn = Re X and Rni P Y for some positive integers n, m and some X, Y E Mod-  R. If / 2^0 for some ideal I of R, then I C -LI. There exists a right ideal J of R maximal with respect to  -L  I fl J^0. Then 1 / ® J < e RR.  JicJnIcJnii= 0. Then J C Jn i / = 0. Hence i / <, RR. Therefore we have (Inn' <,  R m and thus (± fl P <, P. Since PR is strongly  compressible, there exists ZR C E(PR) such that P C Z ( -L i)m fl P. Then  PI C ZI ((± fl P)/ = 0. So PI = 0. Then Pn I 0. It implies that RI = 0. Therefore I = 0. ^ Lemma 3.1.3 Let PR be a progenerator of Mod-R. If PR is strongly compressible, then Z(PR) = 0. In particular, Z(RR) 0. Proof. We can assume that Rrn Pe X for some positive integer rri, and some  X E Mod-R. There exists a submodule N of P such that Z(PR) e N <, P.  Strongly compressible modules^  43  Since P is strongly compressible, there exists YR C E(PR) such that P C Y Z(PR ) El) N. Write Y ED Y2 with Yi  Z(PR ) and Y2 -2 - N as  right R-modules. For each i (1 < i < m), let e i be the element of Rm with ith component 1 and all others 0. Write e i = ai b i for some a i E Yi and b i E Y2 ED X. Since Yl is right singular, a2 <, RR. And e i cri- b i a; L C Y2 El) X ,  for i  1, 2, • • , m. It is easy to see that (e i an R (at)R and  Er e i ct;' is a  direct sum. Noting that PR, and hence RR has finite Goldie dimension by Lemma 3.1.1, we have that dim(Rm) dim(P  e  X) = dim(P) + dim(X) =  dim(Y) dim(X) dim(Yi ) dim(Y2 ) dim(X) = dim(Yi ) X) dim(Yi ) dim(Er  E dim(at)  dim(Y2 e  ee i ct irL) dim(Yi ) + Edim(c i a;L) = dim(Yi )  dim(Y1 )-km•dim(R) = dim(Y1 )-Fdim(Rrn). Thus dim(Yi ) 0,  i.e., Yi = 0. Therefore Z(PR) 0. ^  Lemma 3.1.4 Vategaonkarj. Let R be a semiprime right Goldie ring. Then any submodule of a f.g. free right R-module is compressible. Proof. Since RR is finite-dimensional, there exist f.g.uniform right ideals of R whose sum, say K, is direct and essential in R. By Proposition 1.2, K contains a regular element r of R. Clearly, the map a 1-4 ra, a E R, embeds RR in K. It follows that any f.g. free right R-module can be embedded  in a finite direct sum of f.g. uniform right ideals of R. Then, if M is a submodule of a f.g. free right R-module, there exist f.g. uniform right ideals of R:  h, • • • , In such that M C  1  I  restriction of the obvious map EB: 2_,/, after omitting some of the modules  If M n I, = 0 for some j, then the  ei o ,h embeds M in e io ,h.  Thus,  h and then reindexing, we may assume  that Mn Ii 0 for all i. It follows that M is essential in Let N be an  Strongly compressible modules ^  44  essential submodule of M. Then N n Ii 0 for all i. Since R is semiprime, we have (N n i )/, 0. Thus, tI, 0 for some t E Nn I. Now, consider i  the R-homomorphism f :^N n Ii defined by f (b) tb. If Ker(f) 0, then Ker(f) <, Ii , and so N n^/ / Ker(f) is torsion by Proposition 1.2. i  This is impossible because N n Ii is torsionfree. So f is a monomorphism. Clearly, the map efi^e(N n^provides an embedding of M into N.  ^  Now we can characterize semiprime right Goldie rings as follows. Theorem 3.1.1 The following are equivalent for a ring R: (a) R is semiprime right Goldie; (b) RR is strongly compressible; (c) Every cyclic right ideal of R is strongly compressible; (c') Every cyclic essential right ideal of R is strongly compressible; (d) Every f.g. right ideal of R is strongly compressible; (d') Every f.g. essential right ideal of R is strongly compressible; (e) Every right ideal of R is strongly compressible; (e') Every essential right ideal of R is strongly compressible; (f) Every cyclic submodule of each progenerator of Mod-R is strongly compressible; (f') Every cyclic essential submodule of each progenerator of Mod-R is strongly compressible; (g) Every f.g. submodule of each progenerator of Mod-R is strongly compressible;  Strongly compressible modules^  45  (g') Every f.g. essential submodule of each progenerator of Mod-R is strongly compressible; (h) Every submodule of each progenerator of Mod-R is strongly compressible; (h') Every essential submodule of each progenerator of Mod-R is strongly compressible.  Proof. (e)^(d)^(c)^(b) and (e)^(e')^(d')^(c')^(b). Obviously. (b)^(a). By Lemmas 3.1.1, 3.1.2 and 3.1.3.  (a)^(e). Let I be a right ideal of R and KR < e IR. There exists J C RR such that^0 and I J <, RR. Then K ED J <, RR. By Proposition 1.2, K ® J contains a regular element r of R. Then the map f : (rI)R—÷ IR, which is defined by f(rx) x for all x E I, is an isomorphism. Since KR < e IR, we have (rK)R < e (rI)R by Proposition 1.1. Since R is semiprime  right Goldie, as a submodule of RR, (rI)R is compressible by Lemma 3.1.4. Hence there exists a monomorphism g : (rI) R --+ (rK) R . Since E(I) is an injective module, there exists h : (rK) R E(I), such that h o g f. Since (rK) R <, (rI)R, we have dim(rK) R = dim(rI)R = dim(g(rI)), and thus g(rI) < e (rK) R . Then h is one to one since f is an isomorphism and g is one  to one. Let X = h(rK). Then I = f(rI) h o g(rI) C h(rK) = X C E(I), and XR (rK) R -=1-' KR. Therefore IR is strongly compressible. (h)^(g)^(f)^(b) and (h)^(h')^(g')^(r)^(b). Obviously.  (a)^(h). Suppose that PR is a progenerator of Mod-R, N a submodule of P and KR <e NR. Set S = End(PR). Then we have the Morita  Strongly compressible modules^  46  equivalence F HomR( s PR , --): Mod-R^Mod-S with inverse equivalence G (— — O s P)R: Mod-S^Mod-R. By [2, Prop.21.6], we have  F(K)s <, F(N)s C F(P)s = Ss. We know that the property of being a semiprime right Goldie ring is Morita invariant, and thus S is a semiprime right Goldie ring. By the equivalence of (a) and (e), we have F(N) is a right strongly compressible S-module. Hence there exists Ys C E(F(N)s) such that F(N)s C Ys F(K) s . Then GF(N) C G(Y) GF(K) KR and G(Y) C G(E(F(N))). Noting that F(N)  c F(E(N)) and F(E(N)) is  injective [2, Prop.21.6], we have E(F(N)) C F(E(N)). Hence E(F(N))  F(E(N)), since dimE(F(N)) s = dimF(N) s = dimNR = dimE(N) R dimF(E(N))s < oo by [2, Prop.21.7]. So G(E(F(N)))^GF(E(N) and GF(N)s C G(Y) C GF(E(N)). If rt : GF^lmod-Ris the natural isomorphism, then N C q(G(Y)) C E(N) and 7/(G(Y) tiG(Y) KR. Therefore  N is strongly compressible. ^ Example 3.1.1 An example of a compressible module which is not strongly compressible can be given as follows: Let R be a domain such that R 2 '"=-' R3 as right R-modules. Such a ring R exists by J.D.O'Neill [31]. Clearly RR is compressible, and dim(RR) oo. By Theorem 3.1.1, RR is not strongly compressible. Corollary 3.1.1 The following are equivalent for a ring R: (a) R is semiprime right Goldie; (b) PR is strongly compressible for some progenerator PR of Mod-R; (c) Every cyclic submodule of some progenerator of Mod-R is strongly compressible;  Strongly compressible modules^  47  (c') Every cyclic essential submodule of some progenerator of Mod-R is strongly compressible; (d) Every f.g. submodule of some progenerator of Mod-R is strongly compressible; (d') Every f.g. essential submodule of some progenerator of Mod-R is strongly compressible; (e) Every submodule of some progenerator of Mod-R is strongly compressible; (e') Every essential submodule of some progenerator of Mod-R is strongly compressible.  Proof. By Theorem 3.1.1 and Lemmas 3.1.1, 3.1.2 and 3.1.3. ^ Corollary 3.1.2 A ring R is semiprime right Goldie if and only if R End(Ps), where Ps is a strongly compressible progenerator of Mod-S for some ring S. ^  3.2 Some applications In this section, using the notion of strongly compressible modules we will present many module-theoretic characterizations of semiprime Goldie (prime right Goldie, or prime Goldie) rings, right w-semi-Priifer (right w-Priifer, semi-Priifer, or Priifer) rings as corollaries of Theorem 3.1.1. Theorem 3.2.1 The following are equivalent for a ring R: (a) R is semiprime Goldie; (b) Every f.g. essential submodule of E(R R ) is strongly compressible;  Strongly compressible modules^  48  (c) Every f.g. submodule of E(RR) is strongly compressible; (d) Z(RR ) = 0, and every f.g. nonsingular right R-module is strongly compressible. (e) Every f.g. essential submodule of E(PR ) is strongly compressible for each progenerator P of Mod-R; (f) Every f.g. submodule of E(PR) is strongly compressible for each progenerator P of Mod-R.  Proof. (a)^(d). Clearly Z(RR) = 0. If  MR  is f.g. nonsingular, then M  is embeddable in a f.g. right free R-module by Theorem 1.3. Then M is strongly compressible by Theorem 3.1.1. (d)^(f). This is because for each progenerator P of Mod-R, every f.g.  submodule of E(PR ) is nonsingular when R is right nonsingular. (f)^(e)^(b) and (f)^(c)^(b). Obviously. (b) z (a). By noting that every f.g. essential right ideal of R is essential  in E(R R ), we have that R is a semiprime right Goldie ring by Theorem 3.1.1. It is enough to show that R is left Goldie. Let Q = E(RR). It is well known that Q is a semi-simple Artinian ring and R is a right order of Q. Let x E Q. Then RR < e R+ xR C E(RR). Since R-F xR is essential in E(RR), xR is strongly compressible, and thus there exists Y C E(RR) such that  (R+ xR)R C YR RR. Let y = f -1 (1). Then Y yR and y' = 0, and thus y is a regular element of Q. Write x = yr 1 , 1 = yr 2 for some ri E R. Then x = 7-  0  1  7- 1 . Hence R is also a left order of Q, showing that R is left Goldie.  Strongly compressible modules^  49  Next we show that the characterization theorem of semiprime Goldie rings of LOpez-Permouth, Rizvi and Yousif, which we mentioned in the beginning of this chapter, is a corollary of the previous theorem. To see this we set up a connection between strongly compressible modules and weakly-injective modules which is given by the following proposition. (Comparing it with [26, P rop.3. 7] . ) Proposition 3.2.1 The following are equivalent for an injective right Rmodule E: (a) Every submodule of E is weakly-injective; (b) Every f.g. submodule of E is strongly compressible. Proof. (a)^(b). Let N be a f.g. submodule of E and A an essential submodule of N. Then E(A) = E(N). Since A is weakly-injective, there exists X C E(A) --= E(N) such that N C X --== A. Thus N is strongly compressible. (b)^(a). Suppose that M is a submodule of E. Let A be a f.g. submodule of E(M). Then Mn A is essential in A. Since A is strongly compressible, f  there exists a submodule Y of E(A) such that A C Y -_- M n A. Then f S induces an isomorphism E(Y) E(M n A). Because M n A is essential in A and A C Y C E(A), we have E(Y) = E(A) = E(M n A). There exists B C E(M) such that E(M) = E(A) ® B. If we define g : E(M) --->. E(M) by g(x +b) = f (x) +b for all x E E(A) and b E B, then g is an R-isomorphism and glE(A) = f . Let X = g -1 (M). Since f(A) C M, we have A C X -_=.-_' M and X C E(M). Therefore M is weakly-injective. 0  Strongly compressible modules^  50  Remark 3.2.1 A ring R is called right weakly-semisimple if every right Rmodule is weakly-injective [21]. From the previous proposition, it follows immediately that a ring R is right weakly-semisimple if and only if every f.g. right R-module is strongly compressible.  Corollary 3.2.1 [26, Theorem 3.9]. The following are equivalent for a ring R: (a) R is semiprime Goldie; (b) Every right ideal of R is weakly-injective; (c) Z(RR) = 0 and every nonsingular right R-module is weakly-injective.  Proof. Because the class of weakly-injective modules is closed under taking essential extensions, Proposition 3.2.1 implies that (b) is equivalent to (c) of Theorem 3.2.1, and (c) is equivalent to (d) of Theorem 3.2.1. ^ Proposition 3.2.2 The following are equivalent for a ring R: (a) R is prime right Goldie; (b) Every nonzero cyclic right ideal of R is strongly compressible and faithful; (c) Every nonzero right ideal of R is strongly compressible and faithful; (d) Every nonzero cyclic submodule of each progenerator of Mod-R is strongly compressible and faithful; (e) Every nonzero submodule of each progenerator of Mod-R is strongly compressible and faithful.  Proof. By Theorem 3.1.1.  ^  Strongly compressible modules ^  51  Proposition 3.2.3 The following are equivalent for a ring R: (a) R is prime Goldie; (b) Every f.g. nonzero submodule of E(R R ) is strongly compressible and faithful; (c) Z(R R ) = 0 and every f.g. nonsingular right R-module is strongly compressible and faithful.  Proof. By Theorem 3.2.1. ^ The following are some new characterizations of right w-semi-Priifer (right w-Priifer, semi-Priifer, or Priifer) rings. Proposition 3.2.4 The following are equivalent for a ring R: (a) R is a right w-semi-Prifer ring; (b) Every f.g. essential right ideal of R is a strongly compressible progenerator; (c) Every f.g. essential submodule of each progenerator of Mod-R is a strongly compressible progenerator.  Proof. By Theorem 3.1.1, Proposition 2.1.1, and Proposition 2.1.3. ^ Proposition 3.2.5 The following are equivalent for a ring R: (a) R is a right w-Pritfer ring; (b) Every f.g. nonzero right ideal of R is a strongly compressible progenerator; (c) Every f.g. nonzero submodule of each progenerator of Mod-R is a strongly compressible progenerator.  Proof. By Theorem 3.1.1, and Proposition 2.1.4.  ^  Strongly compressible modules ^  52  Proposition 3.2.6 The following are equivalent for a ring R: (a) R is a semi-Priifer ring; (b) Every f.g. essential submodule of E(RR) is a strongly compressible progenerator; (c) Every f.g. essential submodule of E(P R ) is a strongly compressible progenerator for each progenerator P of Mod-R. Proof. (c)^(b). Obviously. (b)^(a). By Theorem 3.2.1, Proposition 2.1.1, and noting that every f.g. essential right ideal of RR is essential in E(RR). (a)^(c). Let P be a progenerator of Mod-R, and N a f.g. essential submodule of E(PR). By Theorem 3.2.1, N is strongly compressible. Note that NnP <,N. Thus, there exists X C E(N) such that N C  x' --_Nn P. -  Since both N and P have the same finite Goldie dimension, it follows that N can embed in P as an essential submodule. Then N is a progenerator of Mod-R by Proposition 2.1.4. ^  Proposition 3.2.7 The following are equivalent for a ring R: (a) R is a Priifer ring; (b) Every f.g. nonzero submodule of E(RR) is a strongly compressible progenerator; (c) Every f.g. nonzero submodule of E(PR) is a strongly compressible progenerator for each progenerator P of Mod-R; (d) Z(RR ) --= 0 and every f.g. nonsingular right R-module is a strongly compressible progenerator. Proof. (d)^(c)^(b). Clearly.  53  Strongly compressible modules^ (b)^(a). By Theorem 3.2.1 and Proposition 2.1.4.  (a)^(d). By Theorem 3.2.1, Proposition 2.1.9, and the fact that every f.g. nonsingular right R-module can be embedded in a f.g. free R-module.  Proposition 3.2.8 The following are equivalent for a ring R:  (a) R is semi-simple; (b) Every (right) R-module is strongly compressible; (c) Every (right) injective R-module is strongly compressible; (d) E(R R ) is strongly compressible.  Proof. (a)^(b)^(c)^(d). Clearly. (d)^(a). Since every essential submodule of a strongly compressible  module is strongly compressible, it follows from (d) that every f.g. essential submodule of E(RR) is strongly compressible. Then R is semiprime Goldie by Theorem 3.2.1. On the other hand, condition (d) implies easily that R E(RR). Thus R E(R) is semi-simple by [18, Th.4.28].  ^  54  4 Modules over Prater rings Given a ring R, we know that a module MR is projective if and only if  MR is a direct summand of some direct sum of copies of R. Simply from this, we see that there is a special projective module R which determines the structure of all projective modules. For a commutative Priifer domain  R, Fuchs [12] constructed a divisible module 0 with projective dimension at most one which functions as R in the sense that a module MR is divisible with projective dimension at most one if and only if M is a direct summand of some direct sum of copies of 0. In this chapter, we will extend this result to a noncommutative Priifer ring. This work is carried out in Section 2. In Section 1, we establish a structure theorem for modules of projective dimension one over a noncommutative Priifer ring. Besides its own interest, the structure theorem is also needed for the proof of the above-mentioned result.  4.1 Modules of projective dimension at most one First let us recall some concepts in Module Theory. For a fixed module MR,  EXt n (M , —) is the nth right derived functor of Hom(M, —). If 0 -4 A -÷ B ---> . C -+ 0 is a short exact sequence of right R-modules, then we have the long exact sequence in the second variable  0 ---4 Hom(M, A) ---- Hom(M,B) --f I' d^V' 0 --> E xt ° (M , A) --> Ext ° (M, B) --> E xt ° (M , C) --+ Ext i (M, A) -- E xt i (M , B) --->  Modules over Priifer rings^  55  Similarly, Ext"(—, M) is the nth right derived functor of Horn(—, M). And it induces the long exact sequence in the first variable. A basic fact of the Ext functor is that Ext R 1 (M, N) = 0 if and only if any exact sequence 0 N D M 0 splits. The projective dimension of a module MR, denoted by Pd(MR ) or simply by Pd(M), is the smallest nonnegative integer n such that Extn+ 1 (M, N) 0 for all N E Mod-R, if such an integer n exists. If no such n exists, then  Pd(MR ) oo. Also, Pd(MR ) = n if and only if for any projective resolution of MR:  • • • Pn  •  •  •  Pi 4  Po M 0,  Im(4) is projective [22, P90]. Clearly, Pd(MR) = 0 if and only if M is projective. If 0 0 is a short exact sequence of right R-modules with B projective, then, by examining the induced long exact sequence in the second variable, we have Pd(A) Pd(C) — 1.  Lemma 4.1.1 If MR is finitely generated and R is a Pricier ring, then M T(M) ED M/r(M). Proof. Since M/r(M) is f.g. torsionfree, then it is projective by Proposition 2.3.5. Therefore the short exact sequence 0 r(M) M --+ MIT(M) 0 splits, and so M r(M) M/r(M). ^ For some ordinal p, let 0 =M0 CM i C•••CM„C•••CM p .M (a <p) P)  (1)  be a well-ordered ascending chain of submodules of a module MR. The chain (1) is said to be continuous if Mo = j a<0 M, for every limit ordinal /3 < p.  Modules over Priifer rings^  56  Lemma 4.1.2 [Auslander]. For an ordinal p, let 0 = Mo C^C • • • C^C•• • (a < p)^(1) be a well-ordered ascending chain of submodules of a module MR such that (a) ( j a<p Ma = M; (b) (1) is a continuous chain; (c) Pd(M, +1 1M„) < n for some fixed integer n and all 1 < a +1 < p. Then Pd(M) < n. Proof. If n = 0, then, since (b), M,I(U, < ,M,) is projective for all a < p. It follows that Ma (U, < „Mc ) ® Oa for some projective submodule Ma of M. Therefore M Ucy<p Ma = e „pit/a is projective, thus Pd(M) 0. Now assume n > 0. Let Mc, = Mal(U, < ,M,), and Fa be a free right R-module /  mapping onto Ma with kernel i  K. If a is a limit ordinal, then Ma = 0. In this  case we choose 0 as Fa. Therefore we have Pd(K) Pd(MM ) — 1 < n — 1. Let Fa ®, < ,F,'. Since Fa is free, there is a map Ma which lifts the map^By transfinite induction, the map^Ma can be extended to a map Fa^Ma such that if Ka is the kernel, then Ka C Ka for a < a and  lc:, Ka /(u, < „&).  Thus, Pd(Ka /(U, < ,,K,7 )) Pd(Ka) < n — 1.  Note that 0 = K0 C Kl C • • • C AT, C • • is a continuous chain. By the induction hypothesis, Pd(U a<p Ka ) < n — 1. Since U„, <p K, is the kernel of U a<p Fa -4 M, and U, < ,,F0, = 69a<pFa is projective, we obtain Pd(M) < n. i  A module MR is finitely presented if there is an exact sequence 0^K -+ Rn^M^0, where n is a positive integer and K is finitely generated. This is equivalent to the requirement that there exist f.g. modules KR and PR  Modules over PrUfer rings^  57  such that 0 --+K-4P--+M-4 0 is exact (see [2, Ex.17, P233]).  Lemma 4.1.3 Let R be a Priifer ring. A f.g. module MR is finitely presented  if and only if Pd(M) < 1. Proof. Let 0^  0 be an exact sequence with F f.g.  free. If M is finitely presented, then H is finitely generated. Hence H is f.g. torsionfree. By Proposition 2.3.5, H is projective. Therefore Pd(M) < 1. Conversely, if Pd(M) < 1, then H is projective. Since H C F and F has finite Goldie dimension, H is of finite Goldie dimension. By Proposition 2.3.5, H is finitely generated. ^  Lemma 4.1.4 Let R be a Priifer ring, and H a projective submodule of a  torsionfree module FR. If F/H is finitely generated, then F is projective and  F/ H is finitely presented. Proof. Step 1. First we assume R is a Priifer domain (noncommutative), H is free and F/H is f.g. torsion. Write H e){yR: y E Y}. Let Q = Qd(R). Then Q is a division ring. Since F is torsionfree, the map 0: F F®RQ which is defined by 0(x) = x 0 1 is one to one. Since F/H is torsion, and Y is a basis for H, {y 0 1 : y E Y} becomes a basis for the Q-vector space  (FORQ)Q. Suppose FI H=Rd- • • --Fx,„R, where xi = H, i =1,...,m. Clearly F x 1 R + • • • + x n,R + H. For each i, there exists a nonzero r i of  R such that xiri E yiR + • • • + ykR, where k is a fixed positive integer. Let Ho yiR + • • • + ykR, Fo = xiR + • • • + x fli R + Ho . Then the map Fo /Ho F/H defined by (p(e + Ho) -=^H is onto. Claim: Fo 11 H = Ho . Let = xiai + • • • + x m a ni E H, where each a i E R. Write = y2, b1^• • + yi n k for some y ip E Y and 0^bi E R.  Modules over Priifer rings^  58  Then 0 1 = (y i , 0 1)b 1 + • • • + (y,„ 0 1)b n . On the other hand, 0 1 = (x101)ai-F• • •+(x, n 01)a ni = (xiri H-• • •+(x7nrynOl)r; l ani (y1®  nu k for some uk E Q, since each x ri E yiR + • • • + ykR. By  1)u 1 + • • • + (yk 0  i  noting that each b, 0, and {y ®1 : y E Y} is a basis of (F ORQ)Q, we have {Yii • • , Yin}  C  {Yi, • • • yk}. Therefore E Ho. Consequently  Fo n H = Ho,  implying that co is an isomorphism. Since both H o and F0 are f.g. torsionfree, they are f.g. projective by Proposition 2.3.5. Therefore Fo /Ho , and F/H is finitely presented. We note that H o is a direct summand of HR, hence H/Ho is projective. Since F = x 1 R + • • • + x,R+ H and F0 fl H = Ho , we have an R-module isomorphism 0: H1110 Fo (via 0(x + Ho ) x F0 ). Therefore  F/Fo is projective. So we have F Fo e (F/F0 ) and F is projective. Step 2. Assume R is a Priifer domain, H is projective and F/H is f.g. torsion. Then HEX is free for some X E Mod-R. X is, of course, torsionfree. Therefore Fe X is torsionfree and  0.-11 EDX-FEB X--(FEDX)1(111EDX)c-','FIH--0 ,  ,  is exact. Step 1 implies that (F X)/(H  ® X) is finitely presented and Fe  X is projective. Consequently F/H is finitely presented and F is projective. Step 3. We assume R is a Priifer domain, H is projective and F/H is finitely generated. By Lemma 4.1.1, we may assume F/H = (U/H) ED (VIH) where T(F/H) is f.g. torsion, and V/H (F/H)/T(F/H) is f.g. torsionfree. By Proposition 2.3.5, V/H is projective. Therefore we have V '1L'- He(v/H), and so V is projective. In the following short exact sequence:  0 V F FIV (II H --+ 0  Modules over Priifer rings^  59  U/H is f.g. torsion, and V is projective. Therefore Step 2 implies that F is projective and U/H is finitely presented. As a direct sum of two finitely presented modules, F/H is of course finitely presented. Step 4. The general case: R is a Priifer ring. We know that R is Morita equivalent to a Priifer domain S by Theorem 2.3.5. There exists a Morita equivalence G : Mod-R --- Mod-S, and G induces an exact sequence in Mod-S: 0 --* G(H) --+ G(F) —÷ G(F/H) —÷ 0. By [2, Prop.21.6], G(H) s is projective, and G(F/H) s is finitely generated. It is well-known that the singularity of modules is preserved under Morita equivalences (e.g., see [17, P43]). Then the torsionfreeness of FR implies that G(F)s is torsionfree. Therefore Step 3 implies that G(F)s is projective and G(F I H)s is finitely presented. Hence FR is projective and (F/H) R is finitely presented by [2, Ex.11, P262]. ^ A right R-module is called coherent, if every f.g. submodule is finitely presented. Proposition 4.1.1 Every module MR of projective dimension 1 over a Prifer ring R is coherent, and for any submodule N of M, Pd(N) < 1 and Pd(M IN) < 1. Proof. Let N be a f.g. submodule of MR where Pd(M) ----- 1. Then we can write M F/H with F free and H projective. There exists a submodule G of FR such that H CG and N ----= G1H. Clearly G is torsionfree. Therefore N is finitely presented, and G is projective by Lemma 4.1.4. Since 0 -4 G -4 F -+ FIG Ls,' MIN --+ 0 is exact, we have Pd(M/N) < 1. ^  Modules over PrUfer rings^  60  Theorem 4.1.1 Over a Pricier ring, a countably generated right module has projective dimension < 1 if and only if it is the union of a countable ascending chain of finitely presented right modules. Proof. Given a countably generated module MR, then MR is a union of a countable ascending chain of f.g. submodules. If Pd(M) = 0, i.e., M is projective, then every f.g. submodule of M is torsionfree, and hence is projective by Proposition 2.3.5, and hence finitely presented by Lemma 4.1.3. If Pd(M) = 1, then, by Proposition 4.1.1, every f.g. submodule of M is finitely presented. For the converse, we suppose MR is the union of a chain of right finitely presented R-modules: 0 C^C M2 C • • • C  1V17  ,  C•••  Mn By [2, Ex.17, P233], all /V n+1 1/^are finitely presented. Then Lemma 4.1.3 — -  -  implies that Pd(Mn+ i/Mn ) < 1 for all n. By Lemma 4.1.2, Pd(M) < 1.  ^  Let R be a Priifer domain, and 0 H F 4 M 0 be an exact  = e{x R : x E X} is free on X HR = : y E Y}, where the  sequence of right R-modules such that FR and H is projective. By [2, Cor.26.2],  Hg 's are countably generated projective right R-modules. Consider all pairs (Xi, Y) of subsets X i C X, Y C Y such that Fi =  Hi =  ED{Hy :  e{xR : x E Xi} and  y E Y} satisfy Hi = H n Fi . Let i run over an index set I.  Note that H = H, 6 HZ and F, H = where ED{Hy  :yE  Y \ Yz }. Therefore each F, H is projective. Set T = {M, : i E I}, where M2 = (F, H)/ H. Then, clearly, (0), M E  T, and for Mz ,^E T  with  Mi C^(F1 + H)/(F, H) has projective dimension at most  one.  Modules over Priifer rings^  61  Lemma 4.1.5 [Fuchs]. Let R, MR, and T be as above. Then for any count-  able subset A of M, there exists some Mi E 'T with Mi countably generated such that (A) C Mi , where (A) indicates the submodule of M generated by A. Proof. Given a countable subset A of M, there is a countable subset X (1 ) of X such that 4(X( 1 )) contains A. Let Q Q d (R). Then Q is a division ring. Since (X( 1 )) is torsionfree, we have that f : (X( 1 ))^(X(1)) OR Q which is defined by f (a) a 0 1 is one to one. Similarly, g : (X('))  nH  ((X( 1 ))n H)ORQ (g(b) b01) is one to one. Since Q is a flat left R-module, the map /01 : ((X (1) )nH)ORQ^(X(1))ORQ is a monomorphism, where  1 is the inclusion of (X('))  n H into (X( 1 )). Therefore we have the following  commutative diagram:  0 o^(x(1)) n H  ^  1 (X(1))  i ^if Q 1®1} g  0^((X(1)) n H) OR  ^OR Q  Clearly, 1 0 1 is a Q-homomorphism. Since (X(')) is free with a basis X( 1 ),  (X( 1 )) O R Q is a Q-vector space with a basis {x 1 : x E X (1) }. Thus, as a Q-subspace, ((X( 1 )) n H) OR Q has a countable basis which, we may assume,  n H. There is a countable subset y( 1 ) of Y such that all zi E (No /0) Hy . We claim that (X( 1 )) n H c e yey( i ) H In fact, if not, then we can find an h h a + h b E (X (1) ) n H with 0 $ h a E is {z, 0 1 : i E N} with all z i E (X('))  y  .  62  Modules over PrUfer rings^  eye I/v(011y, and hb E e vEy (01-4. But since h01 E “X (1) )11H)ORQ, h®1 = -  E7_ 1 (zi 0 1)qi for some qi E Q. There exist ai E R and c E CR(0) such that qi = ctic -1 for i  1, • • , n. Then hc01 (h 01)c (E 7iL i ziai) ®1. It follows  that he = ElL i ziai. This implies that h a c = 0, contradicting the fact that H is torsionfree. Hence the claim is true. We can select a countable subset  X( 2 ) of X that contains X( 1 ) and satisfies e yEy (i)Hy C (X( 2 )). Repeating this process, we obtain ascending chains of countable subsets X( 1 ) C X( 2 ) C • • • C X(n) C • • • and Y(1) C y( 2 ) C • • • C  y(n)  C•••  of X and Y, respectively, such that (X(n))  n H c e yEy( ) Hy c n  (X(n+ 1 ))  for each n < 1. Let X* =- U n X n , Y* = U n Y n , F* = e{Rx : x E X*}, and (  H*  )  (  )  e{Hy : y E Y*}. Then F* fl H = H*. Thus M* (F* H)I H E  It is clear that (A) C M*, and M* is countably generated. ^ Lemma 4.1.6 Let R, MR, and  T be as above. Given A = (Fi + H)/H E T  and a countable subset A of M, there exists some IVIi = (Fj H)I H E such that (A, Mi ) C Mj, Mi/Mi is countably generated, and Fi C F3 . Note. The required condition Fi C F.; is really indispensable for the proof of the next lemma. Proof. We consider the following short exact sequence: 0 4 (F, H)I^F Fi^(F1 Fi)I[(Fi H)/ Fi] ÷ 0. -  -  Clearly  (FIFi)I[(Fi+ H)1Fi]2='' F (Fi H) -  Modules over Priifer rings^  63  has projective dimension at most one,  F/Fi = ED-PR x E X \ Xi l is free, where  =x+FiEFIFi , and  (Fi + H)/F, (DM Fi )/Fi : y E Y \Y}('-=. 117) is projective with each  (Hy +^Hy countably generated. By Lemma 4.1.5, there exist  X' C X \ Xi , V' C y such that: (a)  EBI{xR x E  X'} n ((Fi + H)/Fi) = 6){(Hy + Fi )/Fi : y E Y'},  (ED{xR : x E x'} n^+ H)) + Fi = ED { Hy Y G^Fi;^(*) and  (b) [ED {±R : x E X'} ((Fi H)/Fi)]/[(FiH)/ Fi] is countably generated; and  (e) [ ® {xR: x E X'} + ((Ft + H)/F2)1/[(Fi H)/Fi]^AF ), where AF is a countable subset of F such that A ={u -FH:u E AF},^E /V}, and AF  {f + [(Fi + 11)1 Fi] : f G OF}.  It is easy to see that condition (c) is equivalent to Es{xR x E X'} Fi + H D E uEAF uR^+ H. Let  ^  Modules over Priifer rings^ F" = El){xR : x E X' U Xi }, H" =^: y E Y' U M" (F" H)/H.  Then, by (*), F" n H = (Ep{xR : x E X'} Fi) n H D (CHy : y E^e^H" .  On the other hand, if ^b  E F"  n H, i.e., b^b2 E H  for some ^b l E El)fxR : x E^b2 E Fi , then b l = b b2 E Ep{xR : x E X i } n (Fi + H).  By (*), b — b 2^a2 for some  a l E WHy : y E^a2 E Fi. Then b — = a2+b2EFinH-= Hi . -  Therefore ^b^(a2 b2) E {Hy : y E^+^ — H".  Consequently we have F" n H H" , and hence M" E T.  Also, by (**), (A, Mi) = (E.EA, uR + H)/H (Fi H)/H H)/H C (F" H)/H M". Clearly Fi C F". Finally M /Mi^[EB4{xR : x E X'} ((f i H)I Fi)]I[(Fi H)I Fi ] H  l  is couritably generated by (b). The proof is complete. 0  64  Modules over Prlifer rings^  65  Now we can prove the following lemma: Lemma 4.1.7 Let R be a &lifer ring and Pd(M) <1. Then there exists a well-ordered continuous chain of submodules 0 = Mo c MI c • • • C Mc, C • • - C Mc, = M (a < p) such that for each a < p, Ma-Fi l Ma is finitely presented. Proof. Step 1. We assume R is a Prfifer domain. Then we can set up  T as in the above discussion. Choose Mo = (0) E T. Suppose we have already chosen all Ma = (Fa + H)/H for all a < a with Mc, E T such that 0 /11„ +1 /Ma is countably generated and F a C Fa+1 for all a + 1 < o . -  (i) a is not a limit ordinal. We are done if M = Ma _ i . If M^Ma-1, then, by Lemma 4.1.6, there exists some Ma E  r  such that M,_ 1 C Ma,  Ma /Ma - 1 is countably generated, and F,_ 1 C Fa . (ii) a is a limit ordinal. We can define Mo. = Lj a<0. Ma. Let Fa =  E a<0. Fa , Ha = EDIHy : y E li a<o. Ya l. Then Fa = 6){xR : x E Ua<cr Xal, and Fa n H = Ha since MI is a chain. Therefore M a = E a< , Mc, --=_(Fa + H)I H E T. Note that 0 Ma+i /M, is countably generated for all a < a. By transfinite induction, we can get a continuous chain of submodules of MR from T: 0 __, Mo c Mi c • • -  • c  Mc, c  • • • c  Mp _--= M  such that M a, d_ i /Ma, is countably generated for all a < p. From the notes before Lemma 4.1.5, each Pd(Ma+i /Ma ) < 1. Then, for each a, Theorem 4.1.1 ensures that there exists a chain of submodules  Modules over Prider rings  ^  Mc, =^C  66  Mc1, C • • C^a+1  such that 4,2 + 1 /4,' is finitely presented for all i. Therefore, without loss of generality, we may assume each Ma+i /Mc, is finitely presented. Step 2. Let R be a Priifer ring. Then, by Theorem 2.3.5, R is Morita  equivalent to a Priifer domain S via an equivalence F : Mod-R Mod-S with inverse G : Mod-S --+ Mod-R. Since Pd(M) < 1, then Pd(F(M)) <1. By Step 1, there exists a continuous chain of submodules of F(M) s : 0= Noc^  c•••  cNa  c•••^  F(M)  such that Na+i /Nc, is finitely presented for all a < p. Since Morita equivalence preserves exactness [2, Prop.21.4], we have G(Ara+ i)/G(N,) ^= G(Na+i Na ). It follows from [2, Ex.11, P262] that G(Na+1 )1G(N,) is finitely presented for each a < p. If Na =  j a< ,  Nu , then G(Na) = ^G(NT) by [2, Prop.21.7].  Therefore we have shown 0 = G(No  ) c  G(Ni)  c • • • c  G(Na )  c • • • c  G(Np ) = GF(M)R  is a continuous chain of submodules of CF(M) R such that G(Na+i )/G(Na ) is finitely presented for all a < p. Since MR GF(M)R, we can get such a similar chain for MR. ^ Now we can prove the main theorem of this section. Theorem 4.1.2 Let MR be a module over a Pricier ring R. Then Pd(M) <  1 if and only if M is the union of a well-ordered continuous chain of submodules 0 = Mo  c^c • • • c^c  • •  • c  Mp M  Modules over Priifer rings^  67  such that Ma+i IX is finitely presented cyclic for all a < p. Proof. The sufficiency follows from Lemmas 4.1.2, 4.1.3. For the necessity, we know that there is a well-ordered continuous chain of submodules 0=Mo CMi C•••CA,C•••CM„=M such that M, +1 /Mc, is finitely presented for all a < p, by Lemma 4.1.7. Therefore, to complete the proof, it suffices to show the fact that for every finitely presented module NR, there exists a finite chain of submodules of NR such that each factor of this chain is finitely presented cyclic. To see this, let N x i R + • • • + x„R be a finitely presented module, and P x 1 R+ • • --Fx,i _ i R. Then N/P is finitely presented cyclic by [2, Ex.17, P233]. If Pd(N) = 0, then P is f.g. torsionfree, and hence projective by Proposition 2.3.5. If Pd(N) = 1, then P is a finitely presented module by Proposition 4.1.1. Therefore P is a finitely presented module with n-1 generators. Thus, the induction hypothesis implies that there is a chain of submodules of P: 0 = Po C C • • • C Pk P such that Pi+i /P, are finitely presented cyclic for all i^0, 1, • • • , k —1. Hence 0 =P0 CP1 C-• •CPk=PCN is the required chain for N ^  4.2 Divisible modules of projective dimension at most one Given a Priifer ring R, we construct a special divisible module 0 with projective dimension at most one by following Fuchs, and then we characterize all divisible right H-modules with projective dimension at most one by using  Modules over Priifer rings  ^  68  the module O.  Lemma 4.2.1 [Fuchs]. Let 0 = Mo C M1 C  • • •  C Ma C • • • ( a < p) be a  well-ordered continuous ascending chain of submodules of MR. Suppose that Ext R1 (Ma+i iMa , X) = 0 for all a +1 < p, and some X E Mod-R. Then Ext R1 (U aK3 Ma , X) = 0 for every P. Proof. We can assume UM a = M. Let 0 -4X-4E-->M-4 0 be an extension of X by M. We want to show that it splits by constructing a module A such that E = X ® A. Let 0^X -4 Ea -4 Ma -+ 0 be the exact sequence induced by the inclusion M a^M. Obviously, this splits for a = 0. Regard E as the union of the ascending chain 0 = Eo C^C  C Ea C • • • (a p) ,  • • •  and suppose that we have found R-submodules A3 of Ed for each < a such that 0 A o C A i C • • • C Ao C • • • (3 < a), is a well-ordered continuous ascending chain satisfying Ef? X ED A, (0 < a). If a is a limit ordinal, then set A a^U fi<a Ap. This will satisfy Ea X 69 A a . If a — 1 exists, then E a til a _ i is an extension of E a _ i /A a _ i^X by Ea /E a _ i^Ma jMa _ i . By our hypothesis, this splits, i.e., E a /A a -1 =--(Ea_i/Aa_i)e(A,,/24,1) for some A a the other hand,  2  A a _ i . Evidently, Ea = X-FA a . On  xnA, = xnEa_ l nA, = xnA, =  0, thus Ea = X ED A a .  Therefore, there is a well-ordered continuous ascending chain 0 Ao  C  Ai C  • •  CA a C • •^< p) ,  Modules over Priifer rings^ such that Ec, X  e  69  A, for all a < p. Set A^Then E = X  e A.  The module a was first constructed by Fuchs. Facchini used a slight  a to study divisible modules over a commutative domain [7]. Here, we follow Facchini for the construction of a. modification of  Given a ring R, for every positive integer k let  Xk^{(r i , • • • , rk) : ri  E  CR(0), i = 1,^, k} and Xo --= {w}•  Set X^j, >0 .Xj . For (r i , • • • , rk), (r;, • • • ,r;), both in X, we define (r i , ••• , r k ) = (r;, • • • ,r;) •#;• k^/ and r i = r: for i^1,•• • , k. Let U be the free right R-module with basis X, i.e., U  = WR (17) [ED(ri)EXI (ri )R] ED [E (r i ,r2 )Ex2 (ri, r2)R] e • • ••  Set  Y =^• • ,rk)rk — (r1, • • • ,rk—i) (ri,• • • ,rk)  E  Xk,k >  (note (p i , • • • , rk_i) w if k 1), and let V be the submodule of U generated by Y. We define  a  An element a of R is called left invertible if ab =1 for some b E R. And such a b is called a right inverse of a. Some basic facts about  a are included  in the following proposition. Proposition 4.2.1 Let ak be the submodule of  a generated by -V + V : E  Ui<k Xi}. Then  ak C • • , and a _ uk>oak• If every element in CR (0) is left invertible, then a^ao if some element in CR(0) is not left invertible, then ak C ak+i for all k; (b)ao wR(ti) w+ v) RR; And alai) is torsion zf R is a right order; (a) 0 c ao Ca l C • • • C  ;  Modules over Priifer rings^  70  (c) For each k > 0, either Ok + 110k 0, or there exists a non-empty subset  Zk Of Xk+i such that a,.^= EDeezk (6 + ak )R with (e ak )-L-^rk +i R, where e^+ V and e = (r i , • • ,r k+i ); (d) Pd(Ok+i I ö k ) < 1 for every k > 0, and Pd(0) < 1; (e) A module DR is called divisible if Dr = D for every r E CR (0). Let DR be a divisible module, and a E D. Then there exists a homomorphism f :a^D with AO = a; (f) If the ring R is an order, then 0 is divisible; (g) Let the ring R be an order. For every divisible module MR, there exists an exact sequence 0 N D M 0 of divisible right R-modules such that D is a direct sum of modules each of which is isomorphic to 01W r i R -  for some ri E CR(0) U {0 }; if M is divisible torsion then we can choose every such r i in CR(0). Proof. (a). Directly from the constructions of a and a k , we have 0 C ao c  c • • • c ak c • • • , and a =uk>oak•  Moreover, w 0 V implies that ao 0.  If r E CR(0) is not left invertible, then, for each k,(ri.,• • • , r k )+V E  ak\ok_i,  where r 1 = • • • = rk = r. If every element in CR(0) is left invertible, then for each (ri, • • • 7 rk) E Xk, (ri, • • • ,rk) V = ((ri, • • • , rk_i) V)sk E where s k is a right inverse of rk. It follows that ak^= • • ••  (b) . For any 0 a E R, wa V. This implies that ao = w R is a free R-module with a single element basis set {0. So Op '-' 7= RR. Since R is a right order,  T(a/ao )  is a submodule. From the construction of 0, we see T(0 laci)  contains a set of generators of  am,.  It follows that OA 7-(a/a0)•  71  Modules over Priifer rings^  (c). If Ok + i/ak 0, then, by (a), CR(0) contains an element which is not left invertible. Set Zk = {(r i , • • • ,r k+i ) E Xk +1 : r k+i is not left invertible }. Note that if e = (r i ,^,rk +i ) E Xk+1 and rk + i is left invertible, then E  ak. Thus e^= 0. From the constructions of 0, ak, and ak+ i, we  have Ok+llak^ENEZk(^ak)R, and for each e = (ri, • • ,rr k+i ) E Zk,  ak )' = rk-FiR. (d). By (c), ak+iiak = EN E zk cx, + ,(+Ok)R. By defining a well-ordering on Zk, we can write ak + ilak as the union of a well-ordered continuous chain of submodules with each factor of the chain isomorphic to some  (  4- + ak)R.  Since 0 --+ r k +1 R -4 R^(e ak)R -÷ o is exact for e = (r1, • • • ,rk+i) E Zk, we have Pd(Ok+i/ak) < 1 by Lemma 4.1.2. Therefore, by Lemma 4.1.2,  Pd(0) < 1. (e). We construct a map q : Uo<kXk^D as follows: Let 71(w)^a. For (r) E Xi , choose one x E D with xr^a and let ii((r)) = x. Suppose for each element e of Xk_i, q(e) has been defined. For (r 1 ,... ,r k ) E Xk, we choose one x E D with xr k = ri((r i , • • • , r k _ i )) and  let 77((r1, • • • , rk) = x. In this manner, we define a map 77 : U o< kXk D. Since U is a free R-module with a basis U o<k Xk , the map q determines uniquely a homomorphism q : U D. From the construction of 77, we see Y C Ker(q), and so V C Ker(q). Therefore there is a natural epimorphism  a U/V UlKer(n). 77 induces a monomorphism UlKer(n) 4 D.  Then  a D is a homomorphism such that (77 o 0)(tv) a. -  (f).  Let Q = Qd(R), e^E i , k ((r ilk ,r i2k ,• • • ,r 21k )^V)a, k E 0, and  t E CR(0). Then taikr i E Q. Write taikt -1 = pZk1 qi k for some q,k E R and  some Pik E CR (0). By [18, Lemma 5.1, P87], pal = r'a i for some ai E R  ^  72  Modules over Priifer rings^  and r E CR(0). Then taikt -1 =--^aiqik, and thus aiqi kt^rtaik. Therefore^((riik, • • • ,r ki k )^V)aik = Ei,k((rik, • • • ,r ikk ,t,r)^V)rtaik = [E i , k ((r iik , • • • ,r ikk ,t,r)^V)a i qik it. (g). Given a divisible module MR. For any nonzero element a in M,  if ar^0 for any r E CR (0), then we let /„ =-- (a , 0)}; otherwise, we set {  /a = {(a, r) : r E CR(0) with ar^0}. For each (a, r) E / a we choose an ,  fa,r E Hom(O/fOrR, M) satisfying fa , r (ti) tiirR) = a. Such an fa , r exists  by (e). Let D =^AC -vla (a flirR). Then Ifa induces a homomorphism  f  fa,r : D  M, and f is clearly onto. Also, f induces an exact  sequence: 0 ---^D^M^0, where N = ker(f).^(2) By (f), D is divisible. To see N is divisible, let x E N and t E CR(0). Since D is divisible, x^yt for some y E D. Let z = f(y) E M. Then zt^f(y)t = f(yt) = f(x) = 0. Therefore the map g : R/tR —+ M  defined by g(b) zb is a well-defined homomorphism. Define h : R^D by h(b) = (Cy ibtR)b. Then h(t) = 0, and thus h induces a homomorphism  h : RItR D. Directly from the definition of D and the map f, we have f oh = g. Let u = y - h(1) E D. Then f(u) = f (y) - f oh(1) = z - g(1) = 0,  and ut yt - h(l)t = x. Therefore N is divisible. The last part of (g) is now clear from the proof above. ^ A short exact sequence of right R-modules: 0^M^M"^0 is called pure if M ' OR L M OR L is a monomorphism for every left Rmodule L. A module NR is called absolutely pure (or FP-injective) if every exact sequence 0 NR MR PR 0 is pure.  Modules over Priifer rings^  73  Proposition 4.2.2 Let MR be a module over a Priifer ring R. Then the  following are equivalent: (a) MR is divisible;  (b) Ext R1 (RIrR,M) = 0, for every r E CR(0); (c) Ext11 (RI I , M) 0, for every f.g. right ideal I of R; (d) MR is absolutely pure.  Proof. (a) 4,;> (b). From the exact sequence 0^rR^R^RI r R^0, we have an exact sequence Hom(R, M)^Hom(rR, M)^Ext l (R/rR, M) 0. Therefore, Ext l (R/rR, M) = 0 if and only if for every homomorphism  -4 M, there exists a homomorphism R^M such that g extends f. If MR is divisible, r E CR(0) and rR 4 M is a homomorphism, then f (r) = yr  rR  for some y E M. Define g : R -+ M by g(1) = y. Then g extends f, and so Ext i (R/rR, M) = 0. Conversely, let x E M, r E CR(0). Clearly f :rR M via Ara). rx is a homomorphism. Since f can be extended  to a homomorphism R 4 M, then x rg(1). Therefore D is divisible. (b)^(c). Trivial. (b)^(c). Let I be a f.g. right ideal of R. From the exact sequence 0 -* IR  4  RR^(RI I)R -+ 0, we have the exact sequence Hom(R, M) 11°772-(± 111)  H om(I , M) ExPR(RI I , M) 0. Therefore ExeR (R//, M) 0 if and  only if Hom(i, M) is onto if and only if each homomorphism f : IR -+ M can be extended to R. We can find a right ideal J of R which is maximal with respect to I fl J = 0. Then I+J--=IEDJis an essential right ideal of R. By Proposition 1.2, I + J contains an element r E CR(0). Write r r 1 r 2 , for some r 1 E I and some r 2 E J, and let K = I + r 2 R = I  e r2R. Obviously  Modules over Priifer rings ^  74  f : I —+ M can be extended to f : K M. K is a f.g. right ideal of R, hence K is projective, since R is semihereditary. By Proposition 1.4, there exist  fa i l  C K and ffi l C Hom(K, R), such that for any a E K, fi (x) = 0 for all  but a finite number of the f i and a = Ea i f,(a). Since K fen(0) 0, there ,  exists, for each i, a qi E Q c /(R) satisfying qi K C R such that fi (a) = qi a for all a E K. For  seKn  CR (0), we have s = Ea i fi (s) = (Ea i qi )s. This  implies that Ea i qi = 1. Since R is also a left order in Q d (R), there exists t E CR(0) such that all tq, E R. Now the divisibility of M implies that we can write f(a,) = xit with all x i E M. Then for any a E K we obtain f(a) = f(Eaiq,a) Ef(ai)(qia) Ex i (tqi )a = xa with x = Ex,tqi E M. Hence the map a xa from R to M is a R-homomorphism that extends f . (c) .4* (d). Megibben and Stenstriim proved, independently, that (c) (d) for an arbitrary ring R (see [30, Prop.1] or [ 35, Prop.2.6]).  ^  The concept of a semicompact module was defined by Matlis in [28], where it was shown that a module over a commutative Prfifer domain is injective if and only if it is divisible and semicompact. The same result holds in a noncommutative Prfifer ring. For a module MR, let R(M) denote the set of subsets of M of the form {x E M : xI = 0} for a right ideal I of R. M will be called semicompact if every finitely solvable set of congruences x x„, (mod Me,) where x„ E M and Mc, E R(M), has a solution in M [28]. If we note a result of StenstrOm [35, Prop.2.5] that an absolutely pure module is injective if and only if it is semicompact, then the following is immediate:  Modules over Priifer rings^  75  Corollary 4.2.1 Let MR be a module over a Pricier ring R. Then M is injective if and only if it is divisible and semicompact. ^ Proposition 4.2.3 Let MR be a module over a Prifer ring R. If Pd(M) = m > 1, then Ext7P(M,D) = 0 for all divisible module DR.  Proof. We induct on m. If m = 1, then, by Theorem 4.1.2, MR is the union of a well-ordered continuous chain of submodules:  0 = Mo c^c • • • C Ma C • • • C M, = M (a < p) such that Ma+i /M„ is finitely presented cyclic for all a < p. Thus, for each a < p, Ma+1 Ma RI la for some f.g. right ideal /R. Since DR is divisible, Proposition 4.2.2 implies that Ext} ? (Ma+i /M„, D) = 0, for every a < p. By Lemma 4.2.1, Ext R1 (M, D) 0. For m > 1, let 0 N R  FR  M 4 0 be an exact sequence with F -  projective. Then Pd(N) = Pd(M)-1 = m-1. Now the induction hypothesis implies that Extlir l (N, D) = 0 for all divisible module DR. From the exact sequence 0 -+ N F M 0, we have ExtMN,D) Ext Rk + 1 (M , D) for all k > 1. Therefore ExtrRn(M,D) = 0 for every divisible module DR. ^ Remark 4.2.1 Proposition 4.2.3 generalizes a result of L.Fuchs [13, Prop.3.9, P126].  We now can give the following characterization of divisible modules of projective dimension at most one: Proposition 4.2.4 Let MR be a module over a Priifer ring R. Then M is divisible with Pd(M) < 1 if and only if it is a summand of a direct sum of modules of the form alwriR, where every ri E CR(0) U {0}.  Modules over Priifer rings^  76  Proof. (=). By Proposition 4.2.1 (g), there exists an exact sequence 0 --X  N^D -+ M -4 0, where N is divisible, and D^@ rEs a I 'Cyr R for a subset S of CR(0) U {0}. If Pd(M) 0, then ExeR (M,N) = 0 from the definition of projective dimension. If Pd(M) = 1, then Proposition 4.2.3 implies Extj(M, N)=-- 0. Hence, 0 -+ N -+ D -+ M - 0 splits. It follows that M is a summand of D. (=). Let D be as above, and M be a summand of D. Then Proposition 4.2.1 (f) implies M is divisible. We know Pd(a) < 1 from (d) of Proposition 4.2.1. Suppose Pd(a) = 1, we have Pd(a IthrR) < 1 for all r E R by Proposition 4.1.1. Then, a similar proof of Proposition 4.2.1 (d) shows that  Pd(D) < 1. If Pd(D) = 1, then we have Pd(M) =-- 1 by Proposition 4.1.1. On the other hand, Pd(D) 0 implies M is projective and hence Pd(M) 0. Therefore Pd(M) < 1 holds if Pd(a) 1. Suppose Pd(a) = 0, i.e., is projective, then  a  a is torsionfree. Therefore 'thrR is f.g. torsionfree. It  follows from Proposition 2.3.5 that fvr R is projective. Therefore we still have Pd(affvrR) < 1. Repeating the argument above, we have Pd(M) < 1.  Corollary 4.2.2 Let MR be a module over a Priifer ring R. Then MR is  divisible torsion with Pd(M) < 1 if and only if it is a summand of a direct sum of modules of the form a lz vr i R, where each ri E CR(0). -  Proof. It follows from the last part of (g) of Proposition 4.2.1 and the proof of Proposition 4.2.4. ^ Let C(R) denote the center of a ring R, and r E CR (0). Suppose 1 s E  C(R) n CR (0). We defin e two maps as follows:  ^  Modules over Priifer rings  77 :  uo<kxk —* a  by 0(w) (r) — (rs)s, and 0((ri, • • • , rk))^(r,ri, • • • ,rk) —^• • • ,rk)s for k > 1.  And  :Uo<kXk^alth-rR  by 0(w) = 0, 0((r)) zb-F wrR, and 0((r i ,r 2 ,  • • • ,  rk))^(r 2 , • • • , rk)^r R if r 1 = r; or 0 if r 1^r.  Then 0 determines uniquely a homomorphism U 4 0, and defines a homomorphism U^OlWrR. It is straightforward to check that Y wrR C Ker(q5) and V C K erect)). Therefore 0 and V, induce canonically two homomorphisms U/(wrR + V) -1+ Ul Ker(0) and U/V Ul Ker(0).  Note that  a/ wrR U/(wrR + V) and a Then the homomorphism 0:  satisfies 0(th + R) = (r) — (rs)s  and  (1) ((r1,• • • ,rk)-F wrR)^(r, r 1 , • • • , rk) — ( rs, r1, • • • , rk)s for k > 1; and the homomorphism : 0 --÷ 0/wrR satisfies WOO = 0, W((r)) =^wrR, and  Modules over Priifer rings^  78  W((r,r2, • • • , rk)) = (r2, • • • i r k )^fOrR, and tlf((r i , r 2 , • • , r k )) = 0 if r i^r.  amo be the natural homomorphism, and (D 0, W induces a homomorphism T : a/a. -+ Let a 4  i  n o (I). Since kli(t7))  i  Lemma 4.2.2 Let 1,W, 0 1 , and W 1 be the same as above. (a) W o = la/trirR• In particular,ItvrR is a summand of 0; -  (b) W 1 0^= la hin-R• In particular, a I ihrR is a .summand of  a/ao .  Proof. (a). Since Z { ktvrR: E UXk} is a set of generators of alzbrR, it -  -  suffices to check that 4/ o fixs every element of Z. However, the verification is straightforward. (b). Similarly. ^  Theorem 4.2.1 If R is a Priifer ring, and C(R) {0, 1} (e.g., if the characteristic of R 2), then MR is divisible with Pd(M) < 1 if and only if it is a summand of a direct sum of copies of  a.  Proof. Note that if R is a Priifer ring, then C(R) C CR(0). Now apply Proposition 4.2.4 and Lemma 4.2.2. ^ Theorem 4.2.2 If R is a Priifer ring, and C(R)^{OM, then MR is divisible torsion with Pd(M) < 1 if and only if it is a summand of a direct sum of copies of  amo •  Proof. By Corollary 4.2.2, and Lemma 4.2.2. ^  79  References [1] J.H.Alajbegovic and N.I.Dubrovin, Noncommutative Priifer rings, J. Algebra 135, 165-176, 1990. [2] F.W.Anderson and K.R.Fuller, Rings and Categories of Modules, Springer-Verlag, New York/Heidelberg/Berlin, 1974. [3] K.Asano, Arithmetische Idealtheorie in nichtkommutativen Ringen, Japanese J. Math. 16, 1-36, 1939. [4] H.Bass, Finitistic dimension and homological generalization of semiprimary rings, Trans. Amer. Math. Soc. 19, 466-488, 1960. 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