N-Strongly Gorenstein Projective, Injective And Flat Modules

Projective modules, injective modules and flat modules are classical topics in algebra. In recent years, the study of Gorenstein projective (injective, flat) modules and strongly Gorenstein projective (injective, flat) modules attracts many authors and many results are obtained.In 2004, Holm discussed the properties of Gorenstein projective (flat) modules related to direct sums and direct summands and proved that the direct products of Gorenstein in-jective modules are Gorenstein injective modules. In 2008, Xiaoyan Yang and Zhongkui Liu discussed the properties of strongly Gorenstein projective (flat) modules under direct sums, direct summands and tensor products, and proved that the direct products of strongly Gorenstein injective modules are strongly Gorenstein injective modules and that strongly Gorenstein flat modules are closed under direct products and direct limit. In 2010, Zhaoyong Huang defined n-strongly Gorenstein modules, and studied the properties of these modules related to direct sums and direct summands.This thesis mainly discuss the closeness of n-strongly Gorenstein projective (injective, flat) modules under tensor products, direct products and direct limit, and the properties of these modules under the change of rings, generalizing some results of Yang and Liu's. We show that n-strongly Gorenstein injective (flat) modules are closed under taking direct limit (direct products), and discuss, for any commutative local noetherian ring (R, m), the proper-ties of n-strongly Gorenstein modules categories under HomR(R,_) and R(?)R_ (denote by R the m-adic completion of a ring R).In addition, this thesis defines the finitely generated properties of n-strongly Gorenstein flat modules, and Puts forward some questions.Denote by P(R) the set of projective modules, I(R) the set of injective modules, F(R) the set of flat modules; SGI(R) the set of strongly Gorenstein injective modules; n-SGP(R) (n-SGI(R),n-SGF(R))the set of n-strongly Gorenstein projective(n-strongly Gorenstein injective,n-strongly Gorenstein flat)modules;mod R the set of finitely generated R-modules, and fik the morphism of injective resolution of Mi.The main results of this thesis are as follows.Theorem 3.1. Let R be a commutative ring and Q a projective R-module.If M∈n-SGP(R),then M(?)Qn-SGP(R).Theorem 3.2. Let R be a commutative ring ring and F a flat R-module.If M∈n-SGF(R), then M(?)F∈n-SGF(R).Theorem 3.4. Let R be left artinian and suppose that the injective envelope of every simple left R-module is finitely generated.If Mi∈n-SGF(R),thenΠMi∈n-SGF(R).Theorem 3.5. Let R be a left noetherian ring.If Mi∈SGI(R) and the direct limit of {Mi} exists,then lim Mi∈SGI(R).Theorem 3.6. Let R be a left noetherian ring.If Mi∈n-SGI(R),and the direct limit of {Mi} and {Imfik},k=1,...,n-1,both exist,then lim Mi∈n-SGI(R).Theorem 4.1. Let(R,m)be a commutative local noetherian ring and M a finitely generated R-module,then (1)M∈n-SGP(R)if and only if M∈n-SGP(R); (2)If R is a projective R-module and M∈n-SGP(R),then M∈n-SGP(R).Theorem 4.2. Let(R,m) be a commutative local noetherian ring,M an R-module,and R a projective R-module.(1)If M∈n-SGI(R),then HomR(R,M)∈n-SGI(R);(2)If HomR(R,M)∈n-SGI(R),then HomR(R,M)∈n-SGI(R).Theorem 4.3. Let(R,m) be a commutative local noetherian ring and M an R-module.(1)If M∈n-SGF(R),then R(?)R M∈n-SGF(R);(2)If R(?)R M∈n-SGF(R),then R(?)R M∈n-SGF(R).Theorem 4.4. Let R be a commutative ring,S a multiplicatively closed subset of R, and S-1 R a projective R-module.(1)If A∈n-SGP(R),then S-1 A∈n-SGP(S-1 R);(2)If S-1 R is a finitely generated R-module,then for any B∈S-1 R-Mod,B∈n-SGP(R) if and only if B∈n-SGP(S-1 R).Theorem 4.5.Let R be a commutative noetherian ring and S a multiplicatively closed subset of R.If B∈mod R∩n-SGP(S-1 R),then B∈n-SGF(R).Theorem 4.6. Let R be a commutative ring,S a multiplicatively closed set of R,and S-1 R a projective R-module.(1)If A∈n-SGI(R),then HomR(S-1 R,A)∈n-SGI(S-1 R);(2)For any B∈R-Mod,HomR(S-1 R,B)∈n-SGI(R) if and only if HomR(S-1 R,B)∈n-SGI(S-1 R).