On Gorenstein Cotorsion Modules

If an exact sequence…â†'F1â†'F0â†'F-1â†'F-2â†'… consists of flat (left) R-modules, i.e., each term Fi (i∈Z) is a flat (left) R-module, and for any injective (right) R-module E. the sequence…â†'E(?)BF1â†'E(?)BF0â†'E(?)BF-1â†'E(?)BF-2â†'… is also exact, then M=ker(F-1â†'F-2) is called a Gorenstein flat (left) R-module. The study of Gorenstein flat R-modules is very extensive and is popular in Gorenstein homological algebra.In [26]. Enochs, E.E. and Lopez-Ramos. J. A. gave the definition of Gorenstein cotorsion modules. A (left) R-module.V is called Gorenstein cotorsion provided that Ext1R(F_V)=0for any Gorenstein flat (left) R-module F. In this dissertation, we in-troduce and study Gorenstein cotorsion dimensions of rings and modules (see Chapter2). Furthermore, we give the definition of FP-Gorenstein modules which is a natural generalizaion of Gorenstein cotorsion modules. A (left) R-module N is called FP-Gorenstein cotorsion provided that Ext1R(F,N)=0for any finitely presented Goren-stein flat (left) R-module F. Some properties of F P-Gorenstein cotorsion R-modules are studied. Similarly. FP-Gorenstein cotorsion dimensions of rings and modules are defined and studied (see Chapter3).The thesis consists of three chapters.In Chapter1. some preliminaries and the main results are given.In Chapter2. some characterizations of Gorenstein cotorsion modules are given. Over a right coherent ring. Gorenstein cotorsion dimension of an R-module is studied. In particular, we compare this dimension with cotorsion dimension of modules. As applicatins, we simplify the proofs of some known results.In Chapter3, FP-Gorenstein cotorsion modules are introduced and some proper-ties of them are given. We discuss the existences of FP-Gorenstein cotorsion precovers and preenvelopes over coherent rings. Furthermore, FP-Gorenstein cotorsion dimen-sions of modules and rings are introduced and we characterize some rings by these dimensions.