Gorenstein Projective Modules And Injective Modules

In this thesis, the properties of Gorenstein projective and injective modules are investigated and many characterizations of modules on QF-rings are given, we obtain that the finitely generated modules on commutative QF-rings are not only projective and injective, but also Gorenstein projective and injective when it satisfies specially designated conditions. Meanwhile, we generalize the mentioned above properties to the direct sums and direct summands. The whole paper is divided into four chapters.In the first chapter, we introduce the background and the value of studying this paper and give some basic definitions and notations of rings and modules.The chapter 2 is the first part of the major content. At the beginning, we give some definitions such as Gorenstein injective modules, injectively resolving class, proper left resolution and coproper right resolution. We conclude that the class of all Gorenstein injective modules is injectively resolving, namely, for a short exact sequence 0→M'→M→M"→0 of left.R-modules, when M'is Gorenstein injective, then M" is Gorenstein injective if and only if M is Gorenstein injective, and the class of all Gorenstein injective modules is closed under arbitrary direct products and direct summands.In chapter 3, we mainly discuss the related properties of Gorenstein injective dimension. Firstly, we define the Gorenstein injective preenvelope of modules and get the following results:For any R-module with finite Gorenstein injective dimen-sion, it admits an injective Gorenstein injective preenvelope; For any R-module M with finite injective dimension, n∈Z+, the following conditions are equivalent:(1) GidRM≤n;(2) ExtRi(E,M)= 0, for all i> n and all R-modules E with idRE<∞;(3) ExtRi(E,M)= 0, for all i> n and all injective R-module E; (4) For every exact sequence 0→M→G0→...→Gn-1→Kn→0, when Gi(i=0, ...,n -1) is Gorenstein injective, then so is Kn;An R- module with finite injective dimension, then the Gorenstein injective dimension and the injective dimension are equivalent.The properties of finitely generated modules on commutative QF-rings are given in chapter 4. Firstly, we characterize QF-rings and present the good qualities of modules which are put on QF-rings. Secondly, the characterizations of finitely generated modules on commutative QF-rings are discussed. We get the following result:Let R be any commutative QF-rings, M be a R- module, if M satisfies the following conditions:(i) pdRM<∞; (ii) End(RM) is a projective R-module. The M is not only projective and injective, but also Gorenstein projective and injective. At last, we generalize the above properties to the direct sums and direct summands.