UBC Theses and Dissertations
Extentions of rings and modules Chew, Kim Lin
The primary objective of this thesis is to present a unified account of the various generalizations of the concept of ring of quotients given by K, Asano (1949), R. E. Johnson (1951), Y. Utumi (1956), G. D. Findlay and J. Lambek (1958). A secondary objective is to investigate how far the commutative localization can be carried over to the noncommutative case. We begin with a formulation of the notion of D-system of right ideals of a ring R. The investigation of the D-systems was motivated by the fact that each maximal right quotient ring of R consists precisely of semi R-homomorphisms into R with domains in a specific D-system of right ideals of R or of R¹, the ring obtained from R by adjoining identity. A nonempty family X of right ideals of R is called a D-system provided the following three conditions holds: D1. Every right ideal of R containing some member of X is in X. D2. For any two right ideals A and B of R belonging to X, φ⁻¹B belongs to X for each R-homomorphism φ of A into R. D3. If A belongs to X and if for each a in A there exists Ba in X, then the ideal sum of aBa (a in A) is in X, Each D-system X of right ideals of R induces a modular closure operation on the lattice L(M) of all submodules of an R-module M and hence gives rise to a set Lx(M) of closed submodules of M. We are able to set up an isomorphism between the lattice of all modular closure operations on L(R) and the lattice of all D-systems of right ideals of R and characterize the D-systems X used in Asano's, Johnson's and Uturai's constructions of«quotient rings in terms of properties of Lx(R). In view of the intimate relation between the rings of quotients of a ring R and the extensions of R-modules, we generalize the concepts of infective R-module, rational and essential extensions of an R-module corresponding to a D-system T of right ideals of R¹. The existence and uniqueness of the maximal Y-essential extension, minimal Y-injective extension and maximal Y-rational extension of an R-module and their mutual relations are established. Finally, we come to the actual constructions of various extensions of rings and modules. The discussions center around the centralizer of a ring over a module, the maximal essential and rational extensions and the different types of rings of right quotients. We include here also a partial, though not satisfactory, solution of the noncommutative localization problem.
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