The Properties Of Strongly Copure Modules And Their Applications

In relative homological algebra, Gorenstein modules and related modules are impor-tant objects, and they have been studied by many authors [6,9,10,14-19,21,28,31]. For example, Enochs and Jenda [14] introduced strongly copure injective modules. Fu, Zhu and Ding [21] introduced strongly copure projective modules. It is easy to see that every Gorenstein injective module is strongly copure injective and every Gorenstein flat mod-ule is strongly copure flat. By [17, Lemma 10.2.4], every finitely presented Gorenstein projective left R-module is strongly copure projective over a left coherent ring R. In this paper, we mainly study strongly copure modules.This paper is divided into four chapters.In Chapter 1, some background and main results are given.In Chapter 2, strongly copure modules and dualizing modules are studied. If there is a dualizing S-R-bimodule associated with a right noetherian ring R and a left noetherian ring S, we study the relations among strongly copure injective left S-modules, Goren-stein injective left S-modules and modules in the Bass class B(S), and the relations among strongly copure projective (flat) left R-modules, Gorenstein projective (flat) left .R-modules and modules in the Auslander class A(R). Then we give the characterizations of strongly copure injective left S-modules, strongly copure projective left R-modules and strongly copure flat left R-modules.In Chapter 3, we study when every module has a strongly copure injective precover and every finitely presented module has strongly copure flat and strongly copure projective preenvelopes. Moreover, we consider the relative derived functors and dimensions.In Chapter 4, homotopy categories of strongly copure modules are studied. We consider when the homotopy category of strongly copure projective modules is compactly generated. Then we give some adjoint functors and recollements in homotopy categories.