n-strongly Gorenstein Projective and Injective and Flat Modules

In this paper, we study some properties of n-strongly Gorenstein projective,injective and flat modules, and discuss some connections between n-strongly Gorenstein injective, projective and flat modules. Some applications are given.

On Rings with Finite Global Gorenstein Dimensions

As a proper setting to study Gorenstein projective and injective dimensions of modules via vanishing of Gorenstein Ext-functors, a notion of a generalized Gorenstein ring is introduced, which is a non-trivial generalization of Gorenstein rings. Moreover, a new proof for Bennis and Mahdou's equality of global Gorenstein dimension is given.

Remarks on Gorenstein Weak Injective and Weak Flat Modules

In this paper,we introduce Gorenstein weak injective and weak flat modules in terms of,respectively,weak injective and weak flat modules;the classes of Gorenstein weak injective and weak flat modules are larger than the classical classes of Gorenstein injective and flat modules.In this new setting,we characterize rings over which all modules are Gorenstein weak injective.Moreover,we discuss the relation between the weak cosyzygy and Gorenstein weak cosyzygy of a module,and also the stability of Gorenstein weak injective modules.

A Note on DG-Gorenstein Injective Complexes

The notion of DG-Gorenstein injective complexes is studied in this article.It is shown that a complex G is DG-Gorenstein injective if and only if G is exact with Z_(n)(G)Gorenstein injective in R-Mod for each n∈Zand any morphism f:E→G is null homotopic whenever E is a DG-injective complex.

Acyclic Complexes and Gorenstein Rings

For a given class of modules A,let A be the class of exact complexes having all cycles in A,and dw(A)the class of complexes with all components in A.Denote by GL the class of Gorenstein injective modules.We prove that the following are equivalent over any ring R:every exact complex of injective modules is totally acyclic;every exact complex of Gorenstein injective modules is in every complex in dw(GL)is dg-Gorenstein injective.The analogous result for complexes of flat and Gorenstein flat modules also holds over arb计rary rings.If the ring is n-perfect for some integer n≥0,the three equivalent statements for flat and Gorenstein flat modules are equivalent with their counterparts for projective and projectively coresolved Gorenstein flat modules.We also prove the following characterization of Gorenstein rings.Let R be a commutative coherent ring;then the following are equivalent:(1)every exact complex of FP-injective modules has all its cycles Ding injective modules;(2)every exact complex of flat modules is F-totally acyclic,and every R-modulc M such that M^(+)is Gorenstein flat is Ding injective;(3)every exact complex of injectives has all its cycles Ding injective modules and every R-module M such that is Gorenstein flat is Ding injective.If R has finite Krull dimension,statements(1)-(3)are equivalent to(4)R is a Gorenstein ring(in the sense of Iwanaga).

Gorenstein Injective and Injective Complete Cohomological Dimensions of Groups

Using Nucinkis＇s injective complete cohomological functors, we assign a numerical invariant to each group P, called the injective complete cohomological dimension of F, denoted by iccd P. We study this dimension and investigate its properties. Also, we define the Gorenstein injective dimension of the group F, which is denoted by Gid F. We show that Gid F is related to iccd F, as well as to spli and silp invariants of Gedrich and Gruenberg. In particular, it is shown that iccd P is a refinement of Gid P. In addition, we show that silp F = spli F 〈 ∞if and only if the Shapiro lemma holds for injective complete cohomology.

Some Remarks on Projective Generators and Injective Cogenerators

In this paper,we give a relationship between projective generators（resp.,injective cogenerators） in the category of R-modules and the counterparts in the category of complexes of R-modules.As a consequence,we get that complexes of W^--Gorenstein modules are actually W-Gorenstein complexes whenever W is a subcategory of R-modules satisfying W⊥W,where W^- is the subcategory of exact complexes with all cycles in W.We also study when all cycles of a W-Gorenstein complexes are W^--Gorenstein modules.