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Radical theory in categorical settings Melvin, Eileen 1973

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c RADICAL THEORY IN CATEGORICAL SETTINGS by Eileen Melvin B.Sc, B.Ed., Memorial University of Newfoundland, 1967 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS -in the Department of Mathematics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1973 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 i t 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 representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Mathematics The University of British Columbia Vancouver 8, Canada Date September , 1973 ABSTRACT The basic theme of the thesis is the development of a theory of radicals in a categorical setting. Guided by the general theory of radicals in rings as presented by Divinsky in his book Rings and Radicals, [1] , we have tried to isolate the minimal categorical assumptions requisite for a "decent" theory of radicals. The primary focus is on the duality between radical and semi-simple classes of objects, and the extendability, or non-extendability, of various properties of the radicals of rings to our two types of categories. In more detail then, the i n i t i a l chapter essentially lists those, and only those, basic categorical propositions that we use. Most of these are in Mitchell, [4], but a few do not seem to have been formulated in the standard texts. In Chapter One, motivated by a paper on radical subcategories by Mary Gray,[2], we have approached the question from the point of view of determining the minimal categorical assumptions underlying the definition of radical. Here we introduce the notion of an A-category, which is rather weaker than that of the motivating, concept - the "semi-abelian" categories of [2], We formulate the notion of a radical property in this setting and show that we can obtain the principal theorem of the Gray paper with these weaker assumptions. Chapter Two contains the most technical results and begins with the introduction of our basic concept - a B-category. This is motivated by Sulinslci's paper on categorical Brown-McCoy radicals, [5]. Guided by the axioms listed there, we have isolated those which seem sufficient for our purposes and proved the most useful properties. We ( i i i ) then show that most of the general radical properties given in Divinsky's book extend to B-categories. Further, we show that in a B-category the notion of radical can be reformulated in such a way that an attractive duality between radical classes and semi-simple classes arises. The question of duality is treated separately in Chapter Three. We show that the construction of upper radical classes is intimately bound to this duality. In the purely ring setting of [1] this duality is rather obscured. The fourth chapter concerns itself with slightly more refined radical properties, again in the setting of a B-category. We examine the notion of "hereditary" properties, and show how the notion of "heart" and its relation to radicals can be extended to our categories. While the thrust of the first four chapters has been theoretical, Chapter Five is quite explicit. Here we construct an example which shows that two of .the major properties of radicals of rings do not extend to B-categories. This example is drawn from the category of finite-dimensional Lie Algebras and is a twenty-one dimensional Lie algebra over the prime field of characteristic three. The motivation for this example comes from an example of Jacobson in "Lie Algebras", [3], With this example our treatment of the general properties of rings as found in Divinsky is complete, for we have been able to decide for essentially a l l the major properties whether or not they are extendable to B-categories. (iv) TABLE OF CONTENTS PAGE Chapter 0 .: Preliminary Definitions and Properties 1 Chapter I : A Categorical Definition of Radical 7 Chapter II : The Concept of Radical in a More Appropriate Categorical Setting . 11 Chapter III : Duality 26 Chapter IV : Hereditary Radical Properties and the Heart of an Object 31 Chapter V : Counterexamples 37 Bibliography '. 48 (v) Acknowledgements I am greatly indebted to Professor L. Roberts for his encouragement and guidance during the preparation of this thesis. The financial support of the University of British Columbia is gratefully acknowledged. Chapter 0 Preliminary Definitions and Properties (1) The following definitions from category theory will be used in the paper, and the properties listed are those which will be needed throughout. Only the last property will be proven. The others are either easy consequences of the definitions or else the proofs are to be found in Mitchell, [4]. Let C be a class of elements (called objects), and for each ordered pair (b,c) of objects of C let C(b,c) be a set. The elements of the set C(b,c) will be called morphisms with domain b and condomain c . We will write a : b c for a e C(b,c) . C is said to be a category i f the following conditions are satisfied: (i) If a : b -»- c and 3 : c -*• d then there is a uniquely defined morphism got : b d. (ii) If a:b-»-c,B:c->-d and y : d g then (y3)a = Y(BO I ) . ( i i i ) For each object b of C there is a morphism 1^  : b -> b such that for any a : c b and any 3 : b -> d we have l^a = a and 31^ =3. 1^ is uniquely defined for each object b and will be called the identity morphism for b . A morphism a : b •* c is an isomorphism i f there exists a morphism a- : c b such that ace' = 1 and a'a = 1, . If there c b exists an isomorphism a : b• -> c we say that b and c are isomorphic objects. (2) A morphism a : b -> c is a monomorphism i f for every pair of morphisms 3, 3' with codomain b such that a3 = ag' we have 3=3' ; i.e. a monomorphism 'is left cancellable. A morphism a : b ->• c is an epimorphism i f for every pair ^ f morphisms 3, 3' with domain c such that Ba = 3'a we have 3=3' ; i.e. an epimorphism is right cancellable. PI) The composition of two isomorphisms is an isomorphism, of two monomorphisms is a monomorphism, and of two epimorphisms is an epimorphism. P2) If get is a monomorphism, then a is a monomorphism - but not necessarily 3 . If 3a is an epimorphism, then 3 is an epimorphism - but not necessarily a . P3) An isomorphism is both a monomorphism and an epimorphism, but the converse is not nesessarily true. An object of C is called a zero object, denoted 0 , i f for every object b in C the sets C(0,b) and C(b,0) each have exactly one element. 0 is unique in the sense that any two zero objects are isomorphic. For every pair of objects b, c the unique morphism b ->• c given by the composition b 0 ->• c will be called a zero morphism, denoted o : b -*• c. P4) The composition of a zero morphism with any other morphism is a zero morphism. P5) o : b -> c is a monomorphism i f f b = 0 . o : b -»• c is an epimorphism i f f c = 0 . (3) Let b be an object of C and consider a l l pairs of the form (£,v) where v : £ •+ b is a monomorphism. Define (£,v) <_ (£}v*) i f there exists a morphism y : £ -> £' such that v'y = v , i.e. the diagram £ —-—> b commutes, i u ! 4-£' If such a y exists i t is a monomorphism since v'y is a monomorphism. Also y is unique since i f v'y = v'n then y = n since v' is a monomorphism. Define an equivalence relation on the pairs (£,v) by (£,v) ^ (£',v') i f f both (£,v) <_ (£>') and UJv') <_ (£,v). We denote the equivalence class containing (£,v) by <£,v> and say that <&,v> is a subobject of b . P 6 ) <£,v> = <b,l^> i f f v : I -> b is an isomorphism. P 7 ) If <£.,v> = <£jv' > then £ and £' are isomorphic objects. The converse is not necessarily true. Note: We will not introduce notation for the dual of the concept of subobject at this point since i t will not be required later. A kernel of a morphism a : b ->• c in a category with zero object is a morphism u : k ->•. b such that (i) ay = o, and (ii) for any morphism u ' : k' b such that ay' = o there exists a unique morphism X : k' -*- k such that yX = y' . A cokernel of a morphism a : b c in a category with zero object is a morphism T : c -> d such that (i) xa = o, and (ii) for any (4) morphism T' : c ->• d' such that T'OE = o there exists a unique morphism n : d -> d' such that nT = x' . P8) A kernel is a monomorphism, and i f u : k -»• b and u1 : k' b are both kernels of a morphism a, then <k,u> = <kju'> . Thus the. sub object <k,y> will be called the kernel of a . We will write kera = <k,u> or simply, kera = u. A cokernel is an epimorphism, and i f T is a cokernel of a then x' is a cokernel of a i f f T = Y t ' where y ^ s an isomorphism. We write coker a = T . P9) If a is a monomorphism then kera = <0,o> If a is an epimorphism then coker a = o . The converses are not necessarily true. P10) If Ba is defined and 3 is a monomorphism, then ker(3a) = ker a, in the sense that i f either exists so does the other and they are equal; in a similar manner, i f a is an. epimorphism then coker (3a) = coker 3. Pli) If a : b ->• c then kera = <b,l^ > i f f a = o ; the cokernel of a is an isomorphism i f f a = o . P12) If u is a kernel and the cokernel of u exists, then u = ker(coker u). If T is a cokernel and the kernel of T exists, then x = coker(ker x). ([4], p.16, Proposition 13.3) . We say that a subobject of an object b is an ideal of b i f i t is the kernel of some morphism with domain b . The image of a morphism a : b c is the smallest subobject of c through which a factors. More precisely the image of a is a (5) subobject <b',v> of c such that (i) there exists a morphism a' : b b* with -va' = a and (ii) i f <£,v'> is any other subobject of c such that there exists a morphism 3 : b -»- £ with v'3 = a then <b',v> <_ <£,y'> . We thus have a unique monomorphism X : b' -»- £ making the following diagram commute: We will denote the object b' by a(b) and write ima = <a(b), v>, or simply ima = v. P13) If v : b -> c is a monomorphism then imv = <b,v> . P14) If T : b -> c is a cokernel then T has an image, namely imt = <c,l > . c Proof of P14: <c,lc> is certainly a subobject of c and T factors as T' = I T . Let v : b' c be any monomorphism such that there exists a morphism T 1 : b -> b' with- V T ' = T . Let T = coker y • y T k y b > c V Now xy = o and so (vx')y= o. But v is a monomorphism so T*y = o . Since T = coker y and T'y = o, there exists a unique morphism n : c b' such that nT = T' . Now . (vn)x = v(nx) = V T ' = T = 1 T (6). and since x is an epimorphism we have vn = 1 . Hence imx = <c,lc> . Because of P15 , when x : b -> c is a cokernel we will sometimes write c = x(b). Note: Throughout the paper we will be concerned with collections of subobjects of an object. Therefore, to avoid logical difficulties, we will consider only categories in which the class of a l l subobjects of an object is a set. Chapter I. A Categorical Definition of Radical Given a category C , we will say that a property ~P is a property of objects of C i f at least one object of C has property "P and, whenever an object has property "P the a l l objects isomorphic to i t also have property 'P . An object will be called a ~P -object i f i t has property ~P . An ideal <k,u> will be called a "P -ideal i f k is a "P -object . Throughout this chapter, we will consider a category C which satisfies the following axioms: Al) C has a zero object, A2) every morphism in C has an image, A3) every kernel in C has a cokernel. For convenience we will call such a category an A-category. Remark: Axioms Al and A2 are weaker that (1) and (2) for a "semi-abelian" category, [2], The following definition is motivated by the definition of a "radical subcategory" in [2]. Definition: Let C be an A-category. We say that a property 1 ? of objects of C is a radical property i f the following conditions are satisfied: Rl) If b is a 1P -object then for every morphism a with domain b , a(b) is a "P -object. R2) For each object b in C there exists a least upper bound in the set of "P -ideals of b , This l.u.b. which is unique will be denoted <r, ,o. > and called the b b radical of b . (8) R3) If b •> c is the cokernel of a, ': r, ->- b then c has b b no non-zero "P -ideals. Remark: Since there is at least one I P -object, Rl ensures that 0 is a -object, and so <0,o> is a f -ideal of every object. R3 then states that <0,o> is the only IP -ideal of the object c , i.e. <r ,c > = <0,o> . c' c A class R of objects of an A-category C will be called a radical class i f the property "is an element of R " is a radical property of objects. We say that <k,u> is an R-ideal If k e R. With respect to a radical property ~P , an object will be called a C F - ) radical object i f i t is a IP -object; that i s , b is a radical object i f and only i f <r ,a^> = <b,l^>. An object will be called ( i P - ) sime-simple i f i t has no non-zero IP -ideals, that i s , b is semi-simple i f and only i f <r^ ,a^ > = <0,o> . Under these conditons, the zero object is the only object that is both radical and semi-simple. Definition: An object c is said to be an extension of an object b by an object d i f there is an exact sequence 0-»-b->c-»-d-*-0. Theorem 1 . 1 : With respect to a radical property, every object of an A-category C is an extension object of -a radical object by a semi-simple object. Proof: Let b be any object of C , <r^ ,o"^ > the radical of b and x : b -> c the cokernel of a, By R2 , r, is a radical (9) object, and by R3 , c is a semi-simple object. Consider the °b T sequence 0 r^ > b > c ->• 0 . Since 0 ->• r is a monomorphism, by P13 its image is <0,o> . Since is a kernel, i t is a monomorphism, and so, by P9 ker = <0,o> . Thus the sequence is exact at r^ . Since 0 is a monomorphism, by P13 im = • Since o^  is a kernel and cokernel exists, by P12 = ker(coker o^) = ker Hence the sequence is exact at b . Since x is a cokernel, by P14 imx = <c,lc>. Also, by Pli , ker(c->0) = <c,lc> . Thus the sequence is exact at c . Remark: Theorem 1.1 is the basic theorem of 2 but in. a weaker categorical setting. . Theorem 1.2: Let R be the class of radical objects with respect to some radical property iP of an A-category. Then for any non-zero object b we have b e R if and only i f for every non-zero morphism a with domain b there exists a non-zero R-ideal of a(b). Proof: If b e R and a is any non-zero morphism with domain b then, by Rl, a(b) c R. Since a is non-zero , a(b) is not the zero object and so <a(b),l N > is a non-zero R-ideal of a(b). a (b ) Assume that for every non-zero morphism a with domain b there exists a non-zero R-ideal of a(b). Let <r, ,a, > be the radical b b of b . By R3 there exists a morphism x with domain b such that x(b) has no non-zero R-ideals, namely x = coker • Hence, because of the assumption, x must be a zero morphism. That i s , coker 0 = 0 (10) and so, by PIL, ker(coker o^) = <b,l^ > . However, by P12, ker(coker o^) = <r^ ,o^ > . Thus <r^ ,o^ > = <h,l^> . That i s , b is a radical object. Theorem 1.3: Let S be the class of a l l semi-simple objects with respect to some radical property 1p of an A-category. Then for any non-zero object b we have b E S i f and only i f for every non-zero ideal <k,u> of b there is a non-zero morphism a with domain k such that a (k)e S. Proof: b e S i f and only i f b has no non-zero ~P -ideals; that is i f and only i f for every non-zero ideal <k,u> of b , k is not a radical object. By Theorem 1.2, k is not a radical object i f and only i f there exists a non-zero morphism a with domain k such that a(k) has no non-zero radical ideals, i.e. such that a(k) is semi-simple. Thus b E S i f and only i f for every non-zero ideal <k,u> of b , there is a non-zero morphism a with domain k such that ct(k) e S . Remark: In Theorem 1.2 we have proven both the necessity condition of Theorem 1, [1], and Lemma 2, [1]. Theorem 1.3 is essentially the necessity condition of Theorem 2, [1]. The sufficiency conditions of Theorems 1 and 2, [1], are the stronger and more useful parts, but these so not hold in our weak categorical setting. In the next chapter we will present a more natural categorical setting for the concept of radical, and there we will prove the f u l l strength of these theorems. Chapter II (11) The Concept of Radical in a More Appropriate Categorical Setting We will now examine the concept of radical in a stronger categorical setting than that of Chapter 1. We want a category with enough structure to permit the development of general radical theory as set out in [1]. Definition; Let <£,v> be a subobject of an object b and let a : b -> c . The image of the morphism av : £ ->- c will be called the image of <&,v> by a . We will sometimes denote the domain of this image by a(£) rather than (av)(£). Let C be a category satisfying .the following axioms: BI) C has a zero object B2) every morphism a factors as a = vx where v ' is the image of a and x is a cokernel. B3) every cokernel has a kernel B 4 ) the image of an ideal by a cokernel is an ideal B4) the set of a l l ideals of an object is a complete lattice i.e. the l.u.b. (or union) of any collection of ideals o an object b exists and is itself an ideal of b ; also the g.l.b.. (or intersection) of any collection of ideals of b exists and is an ideal of, b . For convenience, we will call a category satisfying axioms BI to B5 a"B-category. (12) Lemma 2.1: A B-category is an A-category. Proof: Bl —> Al and B2 ===> A2. We must prove A3 , namel}^  that in a B-category, every kernel has a cokernel. Let <k,u> be any kernel, say u = ker a . By B2 we can write a = VT where v is a monomorphism and T is a cokernel. Thus u = ker vx = ker T . But T is a cokernel so T = coker (ker T), i.e. x = coker u . Hence coker u exists and so every kernel has a cokernel.0 In a B-category we can modify the definition of a radical property of objects as given in Chapter 1. Condition RI can be changed to (RI)' i f b is a f -object and a : b -> c is any cokernel then c is a f -object. RI => RI' since, i f b is a TP-object and a: b •> c is any cokernel then im a = <c,lc> and a(b) = c. Hence by Rl , c is a -object. RI' => Rl since, i f b is a "tP -object and a : b -> c is any morphism then a factors as a = VT where b > a(b) > c and T is a cokernel. By Rl' , <x(b) is a f -object. By axiom B5 , the union of any set of ideals of an object b exists and is itself an ideal of b . Hence the union of a l l iP-ideals of b exists and is an ideal of b . Condition R2 need only specify that this ideal be a lP~-ideal. The definition of a radical property of objects will now be: (13) Definition: A property ~P of objects of a B-category is a radical  property i f the following conditions are satisfied: Rl) If b is a "f -object then for every cokernel a : b -*• c , c is a -object. R2) 1-^  is a "^-object, where <r^ ,a^ > denotes the ideal which is the union of a l l the ^ -ideals of b . <r, .a, > will be called the b b radical of b . R3) If b -* c is the cokernel of a.. : r, -> b then <r ,a > = <0,o>. b b c c Before studying radical properties in B-categories more closely, we will need to examine some of the properties of these categories, and in particular, some properties of cokernels in such a category. Lemma 2.2: In a B-category, every morphism has a kernel. Proof: Let a be any morphism. By B2 , a factors as a = VT where v = im a and T is a cokernel. By B3 , T has a kernel. Since v is a monomorphism and ker x exists, by P10 ker (vx) exists and ker (vx) = ker x . Thus a has a kernel and ker a = ker x . | In general, a morphism that is both a monomorphism and an epimorphism is not necessarily an isomorphism. However, we do have the following: Lemma 2.3: a is an isomorphism i f and only i f a is both a monomorphism and a cokernel. Proof: An isomorphism is a monomorphism and is also a cokernel, namely, by P l i , the cokernel of a zero morphism. (14) If a : b -> c is a monomorphism then by P13, im a = <b,a>. If a : b c is a cokernel then, by P14, im a = <c,lc>. Hence i f a is both a monomorphism and a cokernel then <b,a> = <c,lc>. That i s , by P6, a i s an isomorphism, $ In general, i f a i s a monomorphism then ker a = <0,o> , but not conversely. However, the converse i s true i n a B-category: Lemma 2.4: In a B-category, i f ker a = o then a i s a monomorphism. Proof: By Lemma 2.2, we have ker a = ker x , where a = vx i s the factorization of a given by B2. Since x i s a cokernel, by P12 we have x = coker (ker x) . If ker a = o then ker x = o and so x = coker o . But coker o i s an isomorphism by P l i . Hence a = vx where v i s a monomorphism and x is an isomorphism. Thus a is a monomorphism.? Lemma 2.5: In a B-category, i f Ba i s a cokernel and a is an epimorphism then B i s a cokernel. Proof: By Lemma 2.2, 3 has a kernel, say ker 3 = <£tv>. We w i l l show that B = coker v . Let a : b - * c , 3 : c - * d and ker Bet = <k,u> . Let y : c -> g be any morphism such that yv ,= o . = ker B so there exists (15) a unique morphism 6" : k £ such that v6 = ay. Nov; (ya)u = y( a u) ~ Y ( V ^ ) = (Y V)<5 = o . But pa = coker y , since 3a is a cokernel. Hence there exists a unique morphism 6 : d -»- g such that 8(3a) = Y A • Since a is an epimorphism, 83 = y • Thus for any y : c g such that yv = o there exists a unique morphism 8: d -»- g such that 63=y; i.e. 3= coker v .0 Lemma 2.6: In a B-category, the composition of two cokernels is a cokernel. Proof: Let a.: b c and 3 : c •+ d be cokernels. Since every morphism has a kernel (Lemma 2.2), let <k,y> = ker a and <m,n> = ker3a . We want to show that 3« = cokern • Let Y : b g be any morphism such that Y N = ° « Since n = ker 3a' and (3oOv = 3(ay) = o, there exists a unique morphism 6 : k ->• m such that r\S = y. Since a is a cokernel, a = coker y. But yy = y (r\&) = (Y n)^ =° so there exists a unique morphism x : c g such that x a = Y« Let <a(m),r)'> be the image of <m,n> by . a . By B2 an = n'a' where a' is a cokernel. By B4 , <a.(m),n'> is an ideal of c , since a is a cokernel. Since every kernel has a cokernel, let p = coker n' . (16) Now x(r)'aT) = x(ar\) = (xa)n ~ yr] = o. Since a' is an epimorphism, >;n' "- o . But p = coker n' so there exists a unique morphism 8 : h -> g such that 8p = x« Hence G(pa) = (8p)a = Xa = Y • Now pa : b. ->• h has the properties (i) (pa)n = p(ari) = p(ri'a') = (pn')a' = o, and (ii) for any morphism Y : b ->- g such that yn = o there exists a unique morphism 8 : h -> g such that 6 (pa) = y . Thus pa = coker n . By B2 , Ba factors as Ba = VT where v is the image of Ba and x is a cokernel. But ker x = ker(vx) = ker(Ba) = n so x = coker n . Thus x : b ->- ga (b) and pa : b -* h are both cokernels of n . Hence, by P8 , there exists an isomorphism X : h Ba (b) such that A (pa) = x . m > b 1 > Ba(b) > d Now Ba = vx = v(Apa) , and since a is an epimorphism, B = vAp. Now B is a cokernel, so (vA)p is a cokernel. Also p is a cokernel and hence an epimorphism. Thus, by Lemma 2.5, vA is a cokernel. But vA is a monomorphism since both v and A are monomorphisms. Thus, by Lemma 2.3, vA is an isomorphism. Since pa = coker ri and vA is an isomorphism, by P8 (vA)pa = coker n . That is, Ba = coker n , as required.Q Corollary: If a and 8 are cokernels and <m,n> = ker Ba then the image of <m,n> by a is the kernel of B . (17) Proof: Let <a(m),n'> be the- image of <m,v> by a . As in the proof of Lemma 2.6, (3 = (vX)p where vX is an isomorphism and p = coker n' Hence 3 = coker n' , and since n' is a kernel, <a(m),ri,> = ker 3.$ Lemma 2.7: In a B-category, let 8 :b -f c be a cokernel. If <m,ri> is an ideal of c then there exists an ideal <£,v> of b such that ker 9 <_ <£,v> and <m,ri> is the image of <£ v> by 8 Proof: Let p : c d be the cokernel of n and let <£,v> = ker p8. We will show that ker p8 is the required ideal. k -—> b >c ! > d ^£ m n Let <k,y> = ker 6 . Now (p6)u = p(6u) = o . But v = ker p8 so there exists a unique morphism <5 : k -»• £ such that v6 = u .• Since u is a monomorphism, 6 is a monomorphism. Thus <k,u> <_ <£,v> . By the corollary to Lemma 2.6, since 8 and p are cokernels and <£,v> = ker p8 then the image of <£,v> by 8 is the kernel of p . That is, <m,n> is the image of <£,v> by 8 , as required. S Lemma 2.8: If <k,u> and <£,v> are ideals of an object b such that <k,u> <_ <£,v> then <k,X> is an ideal of £ where X is the monomorphism such that vX = u . Proof: Let <k,u> = ker a and <£,v> = ker 3 • We will show that <k,X> = ker av . (18) (i) (av)A = a(vA) = a y = o. (ii) Let n : m -»- £ be any morphism such that (av)ri = o. Then a(vn) = o. But y = ker a so there exists a unique morphism p : m k such that y p = vn . Hence vAp = vn and since v is a monomorphism, Ap = n • Thus for any n : m •+ £ with (av)n = o there exists a unique p : m k such that Ap = n . By (i) and (ii) , A = ker av and so <k,A> is an ideal of £ .0 The main purpose for Lemmas 2.1 through 2.8 is to enable us to prove the follox^ing theorem which gives an alternative, and very useful, characterization of a radical class of objects in a B-category. Theorem 2.1: Let R be a class of objects of a B-category containing the zero object and satisfying the condition: b e R i f and only i f for every non-zero cokernel with domain b there exists a non-zero R-ideal of a(b). Then R is a radical class of objects. Proof: (Rl) Let b be any object and a : b -> c any cokernel. If a = o then c = 0 and so c e R. Assume a ^  o and suppose c \. R . Then there exists a non-zero cokernel 3 : c -*• d such that d has no non-zero R-ideals. By Lemma 2.6, 3a : b d is a non-zero cokernel. Since d has no non-zero R-ideals, b ij: R. (19) Thus i f b e R and a : b ->• c is any cokernel then c e R. This is condition Rl for a radical class. ( R 2 ) Let <r, ,o\> be the union of a l l the R-ideals of an b b object b . We want to show that r^ e R. If <r, ,o, > = <0,o> then r. =. 0 , and so r, e R. Assume b b b b <r^ ,a^ > f1 <0,o> and let 6 : r^ -> c be any non-zero cokernel with domain r^ . Let <k,u> = ker 0. Since <r^ ,o^ > ^  <0,o> , there exists at least one non-zero R-ideal of b . Now cr^ y is a monomorphism so <k,a^ y> is a subobject of b . If for every non-zero R-ideal <m,n> of b we have <m,n> <_ <k,a^ y> then, by definition of union, <r^ ,a^ > <_ <k,a^ p> . But <k,a^ y> <_ <r^ ,a^ > since y is a monomorphism. Thus <r, ,a, > = <k,a1y> . But this gives b b b <k,y> = < ^ K j l > which is impossible since 6 $ o and <k,y> = ker 8 . rb Hence there exists a non-zero R-ideal of b , say <£,v> such that <£,v> ^ <k,oby> . Since <£,v> is an R-ideal, <&,v> <_ <r^ ,o^ > so there exists a monomorphism p : I -* r^ such that o^p = v . By Lemma 2 . 8 , <£,p> is an ideal of r^ . Let < £ | p ' > be the image of the R-ideal <£,p> by the cokernel 6 . , y 0 k —-—> i- > c D P t J P' 1- >£ By axiom B2 , 8 ' is a cokernel, and so by Rl , which has already been proven, £' is an R-object. By axiom B4 , <S,',p'*> is an ideal of c . (20) Thus <£',p'> is an R-ideal of c . Now i f <£',p'> = <0,o> then Gp = o . Since <k,y> = ker0 this gives <£,p> <_ <k,y> and hence <£,a^p> <_ <k,a^ y>. But <£,o^p> =• <£,v> and <£,v> was chosen so that <£,v> ^_ <k,o^\i>. Thus <£',p'> i= <0,o> . Thus, taking any non-zero cokernel 6 : r^ ->• c we have shown that there exists a non-zero R-ideal of c . Hence r, e R . b (R3) Let 8 : b -*• c be the cokernel of : r^ -»- b . We wish to show that c has no non-zero R-ideals. If 8 = o then c = 0 , so c has no non-zero R-ideals. Let 6 be non-zero and assume <m,n> is a non-zero R-ideal of c . Since 8 is a cokernel and <r^ ,a^ > = ker 8 , by Lemma 2.7 there exists an ideal <£,v> of b such that <r^ ,a^ > <_ <£,v> and. <m,n> is the image of <£,v> by 8 . We will show that <£,v> is an R-ideal of b . By Lemma 2.8, <r^ ,6> is an ideal of £ where 6 is the monomorphism such that v6 = o^  ,,in particular, 6 = ker 0v. But 0v = n6' and n is a monomorphism so 6 = kern0 = ker 0'. Also, since 0' is a cokernel, 8' = coker 6. Now, let X : Z q be a non-zero cokernel and let <k,y> = ker X. (i) If <r, ,5 > <_ kerX then there exists a monomorphism x : rK ^ such (21) that ux = 8 . Then A 6 = A(yx) = (Xy)x = ° » %ut Q ' ~ coker <5 and so there exists a unique morphism T : m ->- q with' x6' = A. By Lemma 2.5, since A and 6' are cokernels, x is a cokernel. x 4 o since A ^  o. Thus we have m e R and x : m q is a non-zero cokernel. Hence, by Rl , q e R and so <q,l > is a non-zero R-ideal of q . (ii) If < r b > ( 5 > jJL berX , let <r^ ,6-'> be the image of < r b » ( S > by A . k __L_> £ _ J _ > • 6 | | 6* A' , rb > r b Then <x^,6t > =f <6,o> , for i f <r^ ,6'> = <0,o> , then A6 = S'A' - o and so <r^ ,6> <_ kerA Now r^ e R , by R2. Since A' : r^ -> r^ is a cokernel, we have r^ e R by Rl . But <r^ ,6'> is an ideal of q by axiom B4 . Hence .<r^ ,<5'> is a non-zero R-ideal of q . Thus for any nonyzero cokernel A : £ q , i.e. either case (i or case ( i i ) , there exists a non-zero R-ideal of q . Hence £ c R and so <£,v> is an R-ideal of b . Thus we have shown that, i f there exists a non-zero R-ideal of c , then there exists a non-zero R-ideal <£,v> of b such that <r^,ob>£> <£,v> and the image of <£,v> by 6 is non-zero. But this is impossible because, i f <£,v> is an R-ideal of b , then <£,v> < <r, ,a,> and since we also have <r, ,o\ > < <£,v> then ' — b' b b b — ' <£,v> = <rb,ab> . Therefore the image of <£,v> by 6 must be <0,o> since Gv = 8ov = o. D (22) Hence i f 6 : b -> c is the cokernel of a, : r, b then b b c has no non-zero R-ideals.8 Remark: Theorem 2.1 is essentially the sufficiency part of Theorem 1, [1]. However, in Theorem 1, [1], the class of rings is assumed to satisfy condition Rl , while here we have used a weaker assumption on the class of objects and have proven Rl . This shows that the f u l l strength of condition Rl is not needed as an assumption. In fact, in a B-category, and in particular, in the category of rings, the f u l l strength of condition Rl is not needed in the definition of a radical property. It can be replaced by the weaker condition (Rl)* - " i f b is a ~P -object then for every non-zero cokernel a : b ->• c there exists a non-zero ~P -ideal of c ". This is proven as a lemma below. We will include the additional assumption that 0 is a ~P -object in order to avoid a "vacuous" type of argument. Lemma: If ~P is a property of objects of a B-category such that 0 is a ~9 -object and "P satisfies conditions (Rl)* , R2 and R3 of a radical property, then iP satisfies Rl . Proof: Assume there exists a non-zero cokernel a : b ->-c such that c is not a "iP -object. Then < r c » ° c > ^ <c'^c> s i n c e > by condition R2 , r is a f object. Thus coker a ^ o . Let 8 : c + d be the c J c cokernel of a c • By R3 , d has no non-zero iP -ideals. Now since a and 8 are both non-zero cokernels, by Lemma 2.6 0a is a non-zero cokernel. Thus there exists a non-zero cokernel 6a : b d and d. has no non-zero 1p -ideals. Hence, by (Rl)* , (23) b is not a ~P -object. Thus we have shown that i f b is a 6 -object then for every non-zero cokernel a : b -> c , c is a "tp -object. If a = o , then c is the zero object and thus a iP -object. Hence we have condition Rl In Chapter I, Theorem 1.2, we showed that any radical class of objects satisfies the condition stated in Theorem 2.1. Thus, in a B-category we have the following alternative definition for a radical class of objects. Definition: Let R be a class of objects of a B-category containing the zero object. We sa}' that R is a radical class of objects i f the following condition is satisfied, b e R i f and only i f for every non-zero cokernel a with domain b there exists a non-zero R-ideal of a(b). The alternative definition for a radical class suggests a method of constructing, for any class of objects N of a B-category, a radical class containing N . This is done for rings in [1] (page 11), and the radical property obtained there is called the lower radical property determined by N . We will give the construction here for a B-category and indicate why the resulting class may be called a "lower" radical class. Given any class N of objects of a B-category we defined a class R^  as follows: (i) Define a class by 0 e and b e N i f and only i f there exists cokernel c b "with c e N. (N^ contains N since 1^  is a cokernel for a l l b in N ). (24) (ii) Let y be any ordinal number and assume that we have defined a class N for every x < y. Define a class N by b e N i f x J J y y and only i f there is some ordinal x < y such that for every non-zero cokernel b -> c there exists a non-zero N ideal of c . x Define by b e i f and only i f b e for some ordinal number y . Then R^  contains N since R^  contains . Lemma: R^  is a radical class of objects . Proof: (i) Assume b e R^  and a : b c is any cokernel. We want to show that c e R^  . If a = o then c = 0 and so c e R^ . Assume a £ o and let _ 8 : c ->• d be any non-zero cokernel. Then 8a : b -»- d is a non-zero cokernel, (Lemma 2.6). Since b e R^  then b e for some ordinal y . Hence there is some ordinal x < y such that there exists a non-zero N ideal of d . Hence c e N and so c e R„ x y as required. (ii) Assume that for every non-zero cokernel a with domain b there exists a non-zero R^-ideal of a(b). We want to show that If a is any non-zero cokernel with domain b then, by assumption, there exists an ordinal x such that a(b) has a non-zero N ideal. Consider the set of ordinals W = {x| there is a non-zero x 1 cokernel a with domain b such that a(b) has a non-zero N -ideal.} ( 2 5 ) Let y be an ordinal number such that y >_ x for a l l x in W . Now for a l l x in W , N £r. N since i f b e N there is x y ' x some x' < x such that for every non-zero cokernel b c there is a non-zero ideal of c . But x' < x implies x' < y and so b e N y Thus there exists an ordinal y such that for every non-zero cokernel b c there is a non zero N ideal of c . Hence b e N y z for z > y and so b e R^  . Thus, by the alternative definition, is a radical class of objects.8 R^  may be called the "lower" radical class determined by N because of the following lemma: Lemma: If R is a radical class of objects of a B-category and N — 'R then £ R . Proof: If c -* b is any cokernel with c e N then, since N •= R and R is a radical class we have b e R. Hence N, £ R . Assume N S R 1 x for a l l ordinals x < y. Let b e N . Then there exists an x < y y such that for every non-zero cokernel b -> c there is an N ideal of c . That Is, for every non-zero cokernel b -»- c there is an R-ideal of c . Since R is a radical class then b e R . Thus N x R for every ordinal x and so R^  S. R . 6 Chapter III ( 2 6 ) Duality In Chapter I, Theorem 1.3, the class of a l l semi-simple objects with respect to some radical property was shown to satisfy a certain condition. In Chapter II, the dual formulation of this condition became an alternative definition for a radical class of objects in a B-category. We are thus motivated to make the following definition: Definition: Let S be a class of objects of a B category containing the zero object. We will say that S is a semi-simple class of objects i f the following condition is satisfied: b e S i f and only i f for every non-zero ideal <k,y> of b there exists, a non-zero cokernel a : k -> c with c e S . Given a radical class R of objects of a B-category define a class ^) (R) by: 0 e s£) (R) and b e (R) i f and only i f for every non-zero ideal <k,u> of b , k | R . Given a semi-simple class S of objects of a B-category define a class ~& (S) by 0 E ^  (S) and b e 1$ (S ) i f and only i f for every non-zero cokernel b c, c £ S . We then have the following theorem: Theorem 3.1: (i) is a semi-simple class (ii) H(S) is a radical class (i i i ) J ("ft (S)) = S (iv) H(J> (R)) = R Proof: (i) b e .29 (R) <==> for every non-zero ideal <k,u> of b , k £ R <=> for every non-zero ideal <k,u> of b , there exists a non-zero cokernel a k c such (27) that for every non-zero ideal <£sv> of c , I \ R <=> for every non-zero ideal <k,u> of b there exists a non-zero cokernel a : k -> c with c e sh (R) . Thus (R) is a semi-simple class of objects. (ii) b e R (S) <==> for every non-zero cokernel b -> c, c | S <==> for every non-zero cokernel b -*• c, there exists a non-zero ideal <£,v> of c such that for every non-zero cokernel a : £ -* d, d £ s <=> for every non-zero cokernel b -> c , there exists a non-zero ideal <£,v> of c such that £ e H (S). Thus (S) is a radical class of objects. ( i i i ) b e.^(~$.(S)) <=*=> for every non-zero ideal <k,u> of b , k 4 TR. (S) <=> for every non-zero ideal <k,u> of b there exists a non-zero cokernel k -> c such that c e S <==> b e S Thus (-fl (S ) ) = S. , (iv) b e "ft (JS (R)) <=«=> for every non-zero cokernel b •> c, c $ J& (R) (28) <=> for every non-zero cokernel b c, c £ A (R) <==> for every non-zero cokernel b c there exists a non-zero ideal <£,v> of c such that I e R <—> b e R . Thus $ (R» = R . B Remark: The preceding theorem and its proof are a play on the duality between the concepts of "radical" and "semi-simple" in a B-category. Its purpose is to emphasize this duality, which is observed in the formulation of Theorem 2, [Ij. Parts (i), (ii) and ( i i i ) of the theorem contain the f u l l strength of Theorem 2, [1], with (i) being simply a rephrasing of Theorem 1 in Chapter I.; Part (iv) does not appear in [1] because, as remarked after Theorem 2.1, the characterization of a radical class in [1] is slightly stronger than necessary, thus upsetting the duality. Immitating the terminology in [1] we can call the class " 1 ^ , (S) the "upper" radical class determined by the semi-simple class S . The following corollary shows why we use the word "upper". Corollary: If S is a semi-simple class and R is a radical class then S S JS) (JR.) i f and only i f R <E "(R (S) . Proof: Assume S c j £ (R) and let b e 1ft (^5 (R)). Then for every non-zero cokernel b -»- c, c £ j& (R). Hence for every non-zero cokernel b •* c , c i S . Thus b e (S), Hence Tre! UJ(R)) £lR.(S) so by part (iv) of Theorem 1.3, R£lft(S). ( 2 9 ) Assume R £ -H (s) and let b e^ S(if? (S)). Then for every non-zero ideal <k,u> of b , k \. iR. (s). Hence for every non-zero ideal <k,u> of b , k \. R. Thus b e £ (R). Hence s$ (1R (s)) £ (R) and by part ( i i i ) of the theorem, S « Je5 (R)..l Thus i f R is any radical class with respect to which a l l the objects of S are semi-simple, i.e. s £r (R) , then R <E. iR, (s), and so (S) may be called the "upper" radical class determined by s • Also i f S is any semi-simple class with respect to which a l l the objects of R are radical, i.e. R £lR(s) , then s £ sS (R). Thus we may also say that JS> (R) is the "upper" semi-simple class determined by R . Remark: Although the concepts of radical class and semi-simple class are -exactly dual in a B-category, we do not expect semi-simple classes to possess the duals of a l l the properties of radical classes because the axioms of a B-category are not self-dual. One particular example of this is that the image of a radical object by a cokernel is radical, while an ideal of a semi-simple object is not necessarily semi-simple. We have already proven the first property and we will give a counterexample to illustrate the second in Chapter V. The example cited is particularly noteworthy because, in the category of rings, the dual statement is true. That an ideal of a semi-simple ring is semi-simple is proven in [1], Theorem 47, corollary 2 . This theorem is the only one of the general radical theorems for rings proved in [1] that does not carry over to a B-category. Not surprisingly, (30) i t is also the only'general' theorem of [ 1 ] whose proof is elementwise rather than structural in that i t relies heavily on the existence and properties of the elements of a ring. Chapter IV (31) Hereditary Radical Properties and.the Heart of an Object Some of the better known radicals in Rings, e.g. the n i l radical, the Jacobson radical and the Brown-McCoy radical, possess the property of being "hereditary". In Chapter 7 of [1] the notion of the "heart" of a ring is used in the study of radicals and, in particular, of hereditary radicals. We can define both of these concepts, that of a "hereditary" radical property and that of the "heart" of an object, in a B-category. subobjects we will state the definition here in order to fix the necessary notation: Definition: Let <k,u> and <£,v> be the subobjects of an object b . The intersection of <k,u> . and <£,v> is a subobject of b , denoted <kn£,t > which satisfies the following conditions: (i) <kn £,VK> ± <k,u> ' and < kn A,t, > <_ <£,v>; i.e. there exist unique monomorphisms k n I k and k n I I , which we will denote by i , and • \ respectively, such that u i , = i , and vx = i , . This (ii) If <m,n> is any other subobject of b such that <m,n> £ <k,u> and <m,n> £ <£,v> , then <m,n> <_ <kA H,IU> . Since these concepts involve the intersection (or g.l.b.) of gives the commutative diagram ik (32) Note: If <k,y> and <Z,v> are subobjects of b such that <k,y> < <£,v> then <k n £ v >' = <k,y> and so <k n £,x > = <k,l > . — b k k Theorem 4.1: For any radical property of objects in a B-category, the following conditions are equivalent: (i) Every ideal of a radical object is radical. (ii) For any object b and any ideal <k,y> of b , we have <k n r, , t, > < <r, ,a, > . b k — k k Proof: Assume that every ideal of a radical object is a radical ideal. Let b be any object and <k,y> any ideal of b . Let <r^,o^> be the radical of b . By axiom B5 , <k,y> n <r^a^> = <k n rb>'lb> i s a n ideal of b. Since <r, ,o, > is an ideal of b and <k n r, , x, > < <r, ,a, >, by b b b b — b b . Lemma 2.8 we have <k n r, , x > is an ideal of r, . But r, is a b r^ b b radical object so, by assumption, k O r^ is a radical object. Since <k,y> is an ideal of b and <k n ^ J 1 ] ^ £. <k, y> , again by Lemma 2.8 we have <k n r, , x, > is an ideal of k . But b k k n r^ is a radical object. Hence <k n r D>ij c > is a radical ideal of k. Now < r ] c » ° ] < - > : l S t n e J - e a s t upper bound of the radical ideals of k . Hence <k n r^, x^ > <_ <r^ ,a^ > , as required. Now assume that for any object b and any ideal <k,y> of b we have <k n ^ J 1 ^ £. < r i c ' P i c > ' ^ e t ^ ^e a radical object. We want to show that, for any ideal <k,y> of b , k is a radical object. Since b is radical, <r, ,o, > = <b,L> . Thus b b ' h (33) <k n r b,i b> = <k,y> n <b,lb> = <k,u> and so <k n r b 5 ' l k > = The assumption then gives <k,l^ > <_ <r-^,a-^' • B u t < rk' C Tk > ^ S a subobiect of k so <r, ,a. > < <k,l, > . Therefore <r, ,a, > = <k,l, > J k' k — ' k v» v > v and hence k is a radical object. k' k k Theorem 4 . 2 : For any radical property of objects in a B-category, the following conditions are equivalent: (i) Every ideal of a semi-simple object is semi-simple, (ii) For any object b and any ideal <k,y> of b , we have < V V < <kn r b,t k>'. Proof: Assume that every ideal of a semi-simple object is semi-simple. Let b be any object and <k,y> any ideal of b . Let <r^,o^> be the radical of b and 6 = coker a. b * -> b k n r. b -> k - r. k -> c -> 6(k) K — > e ' (k) Let y'G' be the factorization of Gy given by axion B2 , i.e. y' is the image of Gy and 6' is a cokernel. By axiom B4 , since 6 is a cokernel, <6(k),u'> is an ideal of c . By R3 , c is a semi-simple object so, by the assumption, G(k) is a semi-simple object. By axion B5 , <k n rb>"lb> i s an ideal of b . Since <k,u> is an ideal of b and <k n r B > ' L B > £. ^ J V ^ > by Lemma 2.8 <k n r K,i v> b' k is. an ideal of k , in particular, <k n r, , i , > = ker Gy . Thus b k (34) <k fi r^, -- ker u'G' = ker 0' since u1 is a monomorphism. Let <r, ,a > be the radical of k and o\'G" the factorization k' k k of given by axiom B2 . Then, since 8' is a cokernel, axiom B4 gives us that <8'(k),o^> is an ideal of 0(k). Since G(k) is semi-simple, 8'(k) is also semi-simple by assumption. But r, is a radical object and 8" is a cokernel. Hence, by Rl , 8'(k) is a radical object. Hence G'(k) must be the zero object since i t is both radical and semi-simple. Since G'(k) is the zero object, = °. B u t <k n r j 3 , i j c > = ker 8' so there exists a monomorphism. 6 : r^ -> k o r^ such that = o^ . . Hence < r k » 0 t c > £ * n rb' Xk > a s r e c L u i r e d -Now assume that, for any ideal <k,u> of an object b , we have <r^,ak> <_ <k n r ^ * 1 ^ • L e t b be a semi-simple object. Then <rb,ob> = <0,o> and so <k n r b , x k > = <0,o> . Thus by assumption, < rk' ak > — < 0 ' o > • B u t < 0 » o > S < ri c» ak > ' H e n c e < rk' ak > = < 0> o > a n d so k is a semi-simple object, as required, B Remark: Theorems 4.1 and 4.2 are Lemmas 68 and 69 of [1] . After Lemma 69, [ l ] 5 i t is stated (but not proven) that, for a given radical property, i f every ideal of a radical ring is radical then every ideal of a semi-simple ring is semi-simple. This does not hold for B-categories in general. In chapter V we give a counterexample to illustrate this. (35) Definition: A radical property of objects in a B-category is said to be hereditary i f for every object b and every ideal <k,u> of b we have <r. ,o\ > = <k n r, , t, > . k k b k Thus Theorems 4.1 and 4.2 state that, with respect to a hereditary radical property we have both (i) an ideal of a radical object is radical and (ii) an ideal of a serai-simple object is semi-simple. Conversely, these two conditions insure that a radical property is hereditary. Definition: The heart of an object b is the intersection of a l l the non-zero ideals of b . Axiom B5 ensures that for every object b in a B-category the heart of b exists and is an ideal of b . Theorem 4.3: With respect to a hereditary radical property the heart of an object is either radical or semi-simple.. Proof: Let <h,p> represent the heart of an object b. If <h,p> = <0 o> then h is both radical and semi-simple. Assume <h,p> f <0,o> . Since <h,p> is an ideal of b and the radical property is hereditary we have <r ,a > = <h n r, ,i,>. Thus <r, ,pa, > = <h n r, ,pi, > = <h n r, , i . > . But <h a r, ,x, > is an h h b ' h b b b b ideal of b and <h n r, ,i,> < <h,p>. Thus <r, ,pa, > is an ideal of b b'b — ' h ' h ( 3 6 ) and <r h ,p0 h > <_ <h,p> . If <r. ,pa. > 4 <0,o> then since i t is an ideal of b we h h have <h,p> <_ <r^,p0^> by definition of the heart. <h,p> . Hence <rj1»PCJ}1> - <b,p> . Since p is a monomorphism this gives <r. ,0, > = <h,l, > . Thus h is a radical object, h' h ' h If <r^,p0 b> = <0,o> then since p is a monomorphism < rh' < 7h > = < 0 » o > ' Thus h is a semi-simple object. 1 Theorem 4.4: Let b be an object with non-zero heart <h,p> . With respect to a hereditary radical property, h is semi-simple i f and only i f b is semi-simple. Proof: With respect to a hereditary radical property, by Theorem 4.2, every ideal of a semi-simple object is semi-simple. Since <h,p> is an ideal of b , i f b is semi-simple then h is semi-simple. Assume h is semi-simple. Then <r^,0^> = <0,o>. Since the radical property is hereditary, < rh> 0j l > - < n n r^, and so <h n r, ,1. > = <0,o>. Now ". b h <h n r b,i h> = <0,o> => . <h n r ^ p i ^ = <0,o> => <h n r b»t b> = <0,o> . If b is not semi-simple then, ^ ^ J 0 ^ f 50>°> a n <^ s o <h,p> _< <rb,ob> by definition of heart. Thus . <h r\ r b,l b> = <h,p> . But <h n rb>'lb> = <0,o> and thus <h,p> = <0,o> . However, by assumption <h,p> 4 <0,o> . Therefore b must be semi-simple. Remark: Theorems 4.3 and 4.4 give us Lemma 74 of [ 1 ] . Chapter V (37) Counterexamples In Chapters III and IV we stated two theorems for rings which, we claimed, do not in general hold for a B-category. These were (i) for any radical property R , every ideal of an R-semi-simple ring is R- semi-simple, and (ii) i f every ideal of an R-radical ring is R-radical then every ideal of an R- semi-simple ring is R-semi-simple. (Of course (ii) would be meaningful only i f (i) were not true). Theorem (i) is a Corollary to Theorem 47, [1] which states that " i f R is any radical property, then for any ring A and any ideal I of A , R(I) is an ideal of A". Thus, by providing a counterexample for (i) we will also have shown that, in a B-category, the radical of an ideal of an object is not necessarily itself an ideal of the object. We find our counterexample for both (i) and (ii) in the category of finite dimensional Lie algebras. Before embarking on its construction we recall some basic facts - see [3] for references. If L is a Lie algebra the derived algebra L' = [L,L] of L is the ideal generated by a l l products [ l ^ l ^ } w i - t n ^ E ^. We then J c- • J • i T (n) T (n-1)' ,T (n-1) T(n-1), T . n define inductively L = L = [L ,L J. L is solvable i f L^n^ = 0 for' some n . It is well known ([3], p. 24) that in any finite dimensional Lie algebra there is a maximal solvable ideal. We call this the radical of L . An easy check verifies that the category of finite dimensional (38) Lie algebras is a B-category and that "L is solvable" is a radical property. We 'recall that L is nilpotent i f and only i f there exists an n such that a l l products of the form [x^[x„[x„ ... [x ., ,x 1 ]...] = 0 , F 1 2 3 n-1' n s and that nilpotent implies solvable. Further we-recall that a Lie algebra is simple if i t has no ideals and i f L' ^  0, It is a basic fact ([3], p.24) that simple implies semi-simple. A complete characterization which we will use of simple, 3-dimensional Lie algebras over fields of characteristic ^  2 is given in [3], p. 13-14. Finally we recall that i f L is any algebra then a derivation of L is an additive map D : L L such that D([a,b]) = [Da,b] + [a,Db] for a l l a,b in L , where [a,b] denotes the multiplication in L . As we are over a field, this implies that D(n) = 0 for a l l n e 2£ . We call Der(L) the set of derivations of L, and note that, under the bracket operation [D^jD^] = D^ T^  ~ °2 D1 ' Der(L) is a Lie algebra. Further general notions will be recalled later. We will construct example of a semi-simple Lie algebra L which contains a non semi-simple ideal I . This is the counterexample for Theorem (i). It.also suffices for Theorem (ii) i f we observe that any subobject of a radical Lie algebra is radical, for L radical means that L^n^ = 0 for some n , and i f I is a subalgebra then T>) <=, L ( n ) - 0 . For example we are forced to consider Lie algebras over fields of non-zero characteristic, since in characteristic zero any (39) ideal of a semi-simple Lie algebra is semi-simple ([3], Ch.III, Cor. 1 to Th. 3j p.73), and the radical of an ideal is the intersection of the ideal with the radical of the containing Lie algebra ([3], Ch. I l l , Th.7, P.24). Our construction will be a modification of the example given on page 75 of [3]. 7L. 2J Let K be the field —; , ( TTZ, is more complex!), the prime field of characteristic 3. Let Z be the commutative K-algebra K. r T 2 3 Z = — . Then dim Z = 3, and 1, x, x (mod x ) form a K-basic, j K. . X xjhile 1, x generate Z as a K-algebra. Hence i f D is a derivation of Z into Z , then D(l) = D(l 2) = 2D(1) so that D(l) = 0 which 2 means that D is K-linear, and since D(x ) = 2xD(x) , D is determined by its value on x . It is easy to check that i f w is an 2 element of Z then setting D (1) =0, D (x) = w, D (x ) = 2xw and W W ' w extending linearly over K makes into a derivation of Z . More succintly, one checks that the map 6 : Z -»• Der Z given by 8(w) = D w establishes an isomorphism of K-vector spaces (even of Z modules, but we won't need that). In particular the derivations D , Dr, D form • x a K-basis of Z . As to the Lie algebra structure of Der(Z) with bracket given by [D^D^ = D ^ - D ^ , we find that [Di_>Dx3 = D i_» ' [D^ ,D 21 = 2D , [D ,D 2 ] = D j . Hence Der(Z) is a three dimensional x X X X X K-Lie algebra which is equal to its own derived algebra. It is well known that such an algebra is simple ([3], p.14). (This is one reason for taking K of characteristic ^ 2). (40) Now let M be the 3-dimensional K-vector space with basis e,f,g and define a bracket operation by setting [e,f] = g, [f ,g] = e , [g,e] = f, and extending linearly. It follows immediately from [3], p. 13-14, that M is a Lie algebra equal to its derived algebra and hence is simple. Let L = M | Z , a K-vector space of dimension 9 , and define a bracket operation by [m^  (3 z^, 8 z^\ - [m^ ,m2] 8 z-^z2 ' T n i s , one can easily verify, defines a Lie algebra structure on ([3], p.30). Finally, set H = L & Der (L) and define a bracket by [(£ 1,D 1), (&2,D2)] = ([£1,£2] +. D 1^2 ~ D 2 £ l ' fD]>D 23)- Z t i s w e l 1 k n o w n that H , the holomorph of L , is then a Lie algebra ([3], p.18). Our claim is that H is a semi-simple Lie algebra and that the canonical image of L in H is a non semi-simple ideal of H . This will then give a l l that we desire. To carry out this program we first investigate L . Let I be 2 2 the subalgebra of L generated by e®x (denoted ex), fx, gx, ex , fx , 2 gx .' Then I is the K vector space generated by these six elements and moreover I is stable under multiplication by e, f, g (eg. [e,fx] = gx etc), hence I is an ideal of L. (I is the Jacobson radical 3 (x) of Z tensored with M.) Since x = 0. in Z i t is immediate that 3 I = 0 , hence I is nilpotent and therefore solvable. One checks that the map M -> ^ / given by m •> m 0 1 mod (I) is an isomorphism of Lie algebras, whence we obtain that ^/ is simple. Since L' = L (verified by an elementary computation) L is not solvable and thus I must be the (41) maximal solvable ideal, that is the radical of L . Now consider the map L -> H. given by I (£,0). By definition of the bracket in H this is a lie. homomorphism and is clearly injective. Moreover since [(&^,0), (^2'^>2^ ~ C-^i'^2^ ~ D2^1'^ ' t^ l G image L of L under this map is an ideal in H which is not semi-simple. In fact I , the image of I , is the radical of L . Hence i t just remains to show that H is semi-simple. First note that i f D is a derivation of Z then 1 8 D : L -> L defined by 1 8 D (m8z) = m 8 Dz is a derivation of L. Further observe that the corresponding map Der (2) Der(L) given by Di—*• 1 8 D is an injective homomorphism of Lie algebras. Hence the composite map Der(Z) ->• H given by D*—$ (0,1 ® D) is also an injective Lie homomorphism. (We will denote (0,1 8 D) by D). Now let R be the radical of H. Then Rn L is a solvable ideal of L and hence is in I. Say 2 2 2 - -m = a^ex + a 2fx + a^gx + a^ex + a,-fx + aggx e R r*\ L £ I . Then [D1,m] = a± [D^exjH^ [D]L,.fx]+a3 [D^gx]+a4[D^ex2]+a5 [D^fx 2]+a 6 [D^gx 2] = a^e + a 2f + a^g + 2ex + 2fx + 2gx , is in I . But since I is the K-vector space with basis ex, fx, gx, ex2, fx 2. gx , this implies that a^ = a^ = a^ = 0 . Applying D^  again to [D^ ,m] we obtain in the same way that a^ = a^ = a^ = 0. Thus R n L = (0) H Hence the canonical projection H -»- / is injective when restricted to L H R and R is isoraorphic to a solvable ideal of / • . (42) Consider then the map Der(L) -*• / given by D>—(0,D) mod(L). L This is clearly a Lie homomorphism which induces an isomorphism of Lie algebras. Hence i t suffices to show that Der(L) is semi-simple. To this end we now let R denote the radical of Der(L) . As is well known, ([3],. p. 9-10), i f a e L then ad a : L L given by (ad a) (x) = [a,x] is a derivation of L , and the map ad : L -»• Der(L) given by a -> ad a is a Lie homomorphism. Further, by definition, the center of L is exactly the kernel of ad. Now we remark that the center of L is zero. This follows from the observation that i f 2 2 2 c = a^e + a^f + a^g + a^ex + a^fx + a^gx + a^ex + a gfx + a^gx is in 2 2 the center then [e,c] = a„g - a„f + acgx - a,fx + aQgx - a nfx = 0 / 3 J o o y which implies that a^ - a^ = a,. = a^ = a g = a^ = 0. Applying f to c then gives that a^ = a^ = a^ = 0. Hence ad is injective and ad(L) is a subalgebra of Der(L) isomorphic to L . Recalling the well known fact ( [3 ] , p.10) that ad(L) , the set of inner derivations i f L , is also an ideal in Der(L) we have that L embeds as an ideal in Der(L) . Now as before we consider R H ad(L). This is a solvable ideal in ad (L) and so must be contained in ad(I). Exactly as before we then consider the effect of 1 0 on an element of R n ad(L) and conclude that Rn ad(L) = (0). Thus, restricted to R , the canonical Der(L) map Der(L) -> — ^ v y is injective and R is isomorphic to a solvable Der(L) ideal of ^ • To complete the argument then we only have to show « u - ' D e r ( 1 ) • • • i that — 5— v — r - is semi-simple, ad (L) We claim that T^ pTjjy ^ s ^ n f a c t simple; more precisely, that (43) that i t i s isomorphic to' Der(Z) . To t h i s end consider the L i e homomorphism Der(Z) ->• SfLL i^.) given by D > — 1 0 D mod (ad(L)). I t ad (L) s u f f i c e s to show, i n view of the s i m p l i c i t y of Der(Z), that t h i s i s an isomorphism. This w i l l require however, a b i t of work. F i r s t we compute the e f f e c t of ad z for z running over 2 2 2 the basis e,f,g,ex,fx,gx,ex ,fx ,gx of L . Taking t h i s ordering of the basis and w r i t i n g the e f f e c t of ad z i n columns we see that, . for example, ad e has the matrix: ad e = while ad ex and ad ex = 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 o' 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 ( 4 4 ) Similarly with f and g , but working from the basic units for f and 0 -1 0 1 0 0 0 0 0 0 0 1 0 0 0 -1 0 0 for g. Adding these we find that the general element of ad (L) has the form: 0 "a3 a2 0 0 0 0 0 0 a3 0 _ a l 0 0 0 0 0 0 a2 a l 0 0 0 0 0 0 0 0 ~a6 a5 0 " a3 a2 0 0 0 a6 0 a3 0 " a l 0 0 0 a 5 a4 0 "a2 a l 0 0 0 0 0 a8 0 a5 0 " a3 a a9 0 "a7 a6 0 ~a4 a3 0 -a a8 a7 0 " a5 a4 0 "a2 a l 0 Note that this implies that i f D e ad (L) is such that i f any two of D(e), D(f), D(g) are zero then D = 0 . In a similar way we can calculate the matrices for the "outer derivations" 1 8 D^ , 1 0 D^  and 1 0 D ^ a n ^ adding these we find x that the image of Der (Z) in Der (L) consists of a l l those matrices, with respect to the above basis, of the form 0 . 0 0 a i o 0 0 0 0 0 0 0 0 0 a i o 0 0 0 0 0 0 0 0 0 a10 0 0 0 0 0 0 a l l 0 0 2 a i o 0 0 0 0 0 0 a l l 0 0 2 a10 0 0 0 0 0 0 a l l 0 0 2 a i o 0 0 0 a12 0 0 2 a l l 0 0 0 0 0 0 a12 0 0 2 a n 0 0 0 0 0 0 a12 0 0 2 a n ( 4 5 ) Note that the image of Der (Z) in Der (L) has the property that each of its elements k i l l s e,' f and g . This implies that the map Der (Z) -> ~ | r ^ ^ is injective. We are then reduced to showing that i t is subjective. We do this by establishing two facts: (i) V = {D E Der (L) | D(e) = D(f) = D(g) = 0} has dimension 3 over K and (ii) V + ad (L) = Der (L) , (as K vector spaces). Then we are done since the remarks above imply that the image of-Der (Z) = y-, whence Im Der (Z) ©• ad (L) = Der (L) as K vector spaces and Der (Z) ->- ^ r - ^ ^ is an isomorphism. For (i) , since the image of Der (Z) C y we know that dim^ V >_ 3 . Thus i t sxiffices to show that dirn^ V <_ 3 . Let D e V . 2 2 2 Then since the elements ex , fx , gx are products of the elements ex, fx and gx , D is determined by its effect on ex, fx, gx . But since fx = [g,ex] and gx = [e,fx] = [e,[g,ex]], D is known i f D(ex) is known. Now D(ex) = D([f,gx]) = [Df,gx] + [f,D(gx)j = [f,D(gx)] = [f,D([e,fx])] = [f,[De,fx] + [e,D(fx)j] = [f,[e,D(fx)] = If,[e,D([g,exJ)]] = [f,[e,[Dg,ex] H-..[g,D(ex)']"]] = [f,[e,[g,D(ex)]]] (!) (46) 2 2 2 So set D(ex) = a^e + a^f + a^g + a^ex + a^fx + a^gx + a^ex + a gfx + a^gx . Then 2 2 D(ex) = [f,[e,[g,D(ex)]]] = [f,[e,a.jf - a2e + a^fx - a5ex + a 7fx - agex ]] 2 = [f ^ g + a4gx + a?gx ] 2 = a^ e + a^ex + a_,ex , .• • . . 2 so that a_ = a„ = a r = a, = a_ = an = 0 .. Whence D(ex) = a,e + a,ex + a7ex and card V <_ 27 , so that dirn^ V <_• 3 and (i) is done. To show (ii) requires a bit more analysis, though of a purely elementary linear algebra character. Our approach is as follows: i f D is any derivation of L we will show that the first three columns of the matrix of D have precisely the form of the first three columns of an element of ad (L) . This implies that there, is a d e L such that the matrix of D - ad (d) has a l l first three columns zero, i.e. that D - ad (d) k i l l s e, f and g. But by (i) this means that D - ad (d) =10 Drr for some w e Z, so that D = ad (d) + 1 0 D and W w we will be done. To show that the columns of D are as above we consider: 2 2 2 , D(e) = a-^ e + a 2f + a^g + a^ex + a,, fx• + a&gx + a?ex + a gfx + aggx 2 2 2 D(f) = b e + b 2f + b 3g + b/+ex + b $fx + b&gx + b?ex + b gfx + bggx 2 2 2 D(g) = c 1e + c 2f + c 3g + c^ex + c 5fx + cggx + c?ex +. c gfx + cggx . and use the facts that D(g) = D(ef) =• [De.f] + [e,Df] D(f) = D(ge) = [Dg,e] + [g,De] D(e) = D(fg) = [Df,g] + [g.Df] . (47) Writing these out in terms of the basis elements and equating coefficients leaves us with a very simple set of linear equations which (using the characteristic of K 2!) simply say that the first three columns of the matrix of D are exactly like those of an element of ad(L) . Q.E.D. 21 Remark: Our example, which we think is minimal, has 3 elements! (48) Bibliography Divinsky, N., Rings and Radicals, University of Toronto Press, Toronto, 1965. Gray, M. W.,"Radical Subcategories," Pacific Journal of Mathematics, 23 (1967), pp. 79-89. Jacobson, N., Lie Algebras, Interscience, John Wiley & Sons, New York, 1962. Mitchell, B., Theory of Categories, Academic Press, New York, 1965. Sulinski, A., "The Brown-McCoy radical in categories," Fundamenta Mathematicae, 59 (1966), pp. 23-41. 

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