Gorenstein Projective, Injective And Flat Modules

In this dissertation,we mainly study some properties and extensions of Gorenstein projective,injective and flat modules.The whole thesis is made up of four chapters.In Chapter 1,we give some necessary conditions for rings under which the direct limits of Gorenstein projective and injective modules is Gorenstein projective and injective respectively.We also study some connections between Gorenstein injective and Gorenstein flat modules and give some characterizations of all modules are Gorenstein flat modules.In Chapter 2,the strongly Gorenstein homological properties of modules are studied.We prove that,under some additional conditions,the class of all strongly Gorenstein projective R-modules is projectively resolving and closed under direct summands.Simultaneously,we prove that,under some additional conditions,(1) R-module M is strongly Gorenstein flat if and only if M~+ is strongly Gorenstein injective;(2) R-module M is strongly Gorenstein injective if and only if M~+ is strongly Gorenstein flat.In Chapter 3,the V-Gorenstein homological properties of modules are investigated.We first introduce the notion of V-Gorenstein flat module and give some characterizations of V-Gorenstein flat module.Secondly, we note only prove that(V-GF,V-GF~⊥) is a perfect hereditary cotorsion theory under some categories,where V-GF denotes the class of all V-Gorenstein flat left R-modules.But also,we show that every left R-module has a V-Gorenstein flat preenvelope.Thirdly,we give some properties of V-Gorenstein projective modules and V-Gorenstein injective modules.Simultaneously,we study the existence of V-Gorenstein injective cover and preenvelope.By analogy with the projective,injective and flat modules,we consider some properties and connections of V-Gorenstein projective,injective and flat modules.In Chapter 4,the C-Gorenstein homological properties of modules are discussed.A necessary condition for rings under which the inverse limits of C-Gorenstein injective modules is C-Gorenstein injective is gived.We also give a characterization of module with finite C-Gorenstein projective dimension.Secondly,we study some interior connections between C-Gorenstein injective and C-Gorenstein flat modules.Simultaneously,we consider these properties under extensions and change of rings.Specifically,we consider completions of rings,Morita equivalences,(almost) excellent extensions,polynomial extensions and localizations and so on.