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.

A Generalization of Gorenstein Injective and Flat Modules

In this article, we introduce and study the concept of n-Gorenstein injective （resp., n-Gorenstein flat） modules as a nontrivial generalization of Gorenstein injective （resp., Gorenstein flat） modules. We investigate the properties of these modules in various ways. For example, we show that the class of n-Gorenstein injective （resp., n -Gorenstein flat） modules is closed under direct sums and direct products for n ≥ 2. To this end, we first introduce and study the notions of n-injective modules and n-flat modules.

Generalized Gorenstein Modules

We introduce a generalization of the Gorenstein injective modules:the Gorenstein FPn-injective modules(denoted by GI_(n)).They are the cycles of the exact complexes of injective modules that remain exact when we apply a functor Hom(A,-),with A any FP_(n)-injective module.Thus,GL_(o)is the class of classical Gorenstein injective modules,and GI_(1)is the class of Ding injective modules.We prove that over any ring R,for any n≥2,the class GI_(n)is the right half of a perfect cotorsion pair,and therefore it is an enveloping class.For n=1 we show that GI_(1)(i.e.,the Ding injectives)forms the right half of a hereditary cotorsion pair.If moreover the ring R is coherent,then the Ding injective modules form an enveloping class.We also define the dual notion,that of Gorenstein FP_(n)-projectives(denoted by GP_(n)).They generalize the Ding projective modules,and so,the Gorenstein projective modules.We prove that for any n≥2 the class GP_(n)is the left half of a complete hereditary cotorsion pair,and therefore it is special precovering.

Gorenstein n-X-Injective and n-X-Flat Modules with Respect to a Special Finitely Presented Module

Let R be a ring,X a class of R-modules and n≥1 an integer.We intro-duce the concepts of Gorenstein n-X-injective and n-X-flat modules via special finitely presented modules.Besides,we obtain some equivalent properties of these modules on n-X-coherent rings.Then we investigate the relations among Gorenstein n-X-injective,n-X-flat,injective and fiat modules on X-FC-rings(i.e.,self n-X-injective and n-X-coherent rings).Several known results are generalized to this new context.

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.

Algebraic K-theory of Gorenstein projective modules

We introduce the Gorenstein algebraic K-theory space and the Gorenstein algebraic K-group of a ring, and show the relation with the classical algebraic K-theory space, and also show the ＇resolution theorem＇ in this context due to Quillen. We characterize the Gorenstein algebraic K-groups by two different Mgebraic K-groups and by the idempotent completeness of the Gorenstein singularity category of the ring. We compute the Gorenstein algebraic K-groups along a recollement of the bounded Gorenstein derived categories of CM-finite Gorenstein algebras.

Gorenstein AC-Projective and AC-Injective Modules over Formal Triangular Matrix Rings

Let A and B be rings and U a(B,A)-bimodule.If BU is flat and UA is finitely generated projective(resp.,BU is finitely generated projective and UA is flat),then the characterizations of level modules and Gorenstein AC-projective modules(resp.,absolutely clean modules and Gorenstein AC-injective modules)over the formal triangular matrix ring T=(A0 UB)are given.As applications,it is proved that every Gorenstein AC-projective left T-module is projective if and only if each Gorenstein AC-projective left A-module and B-module is projective,and every Gorenstein AC-injective left T-module is injective if and only if each Gorenstein AC-injective left A-module and B-module is injective.Moreover,Gorenstein AC-projective and AC-injective dimensions over the formal triangular matrix ring T are studied.

Gorenstein FP∞-Injective Modules and w-Noetherian Ring

We study some homological properties of Gorenstein FP∞-injective modules,and we prove(1)a ring R is not necessarily coherent if every Gorenstein FP∞-injective R-module is injective,and(2)a ring R is not necessarily coherent if every Gorenstein injective R-module is injective.In addition,we characterize w-Noetherian rings in terms of Gorenstein FP∞-injective modules,and we prove that a ring R is w-Noetherian if and only if every GV-torsion-free FP∞-injective R-module is Gorenstein FP∞-injective,if and only if any direct sum of GV-torsion-free FP∞-injective R-modules is Gorenstein FP∞-injective.

The stable Auslander-Reiten components of certain monomorphism categories

Let Λ be an Artin algebra and let Gprj-Λ denote the class of all the finitely generated Gorenstein projective Λ-modules. In this paper, we study the components of the stable Auslander-Reiten quiver of a certain subcategory of the monomorphism category S(Gprj-Λ) containing boundary vertices. We describe the shape of such components. It is shown that certain components are linked to the orbits of an auto-equivalence on the stable category Gprj. In particular, for the finite components, we show that under certain mild conditions,their cardinalities are divisible by 3. We see that this three-periodicity phenomenon reoccurs several times in the paper.

Gorenstein Projective Dimensions of Complexes

We show that over a right coherent left perfect ring R, a complex C of left R-modules is Gorenstein projective if and only if C^m is Gorenstein projective in R-Mod for all m E Z. Basing on this we show that if R is a right coherent left perfect ring then Gpd（C） = sup{Gpd（C^m）｜m ∈ Z} where Gpd（-） denotes Gorenstein projective dimension.

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.

Strongly Gorenstein Flat Dimensions

This article is concerned with the strongly Gorenstein flat dimensions of modules and rings.We show this dimension has nice properties when the ring is coherent,and extend the well-known Hilbert＇s syzygy theorem to the strongly Gorenstein flat dimensions of rings.Also,we investigate the strongly Gorenstein flat dimensions of direct products of rings and（almost）excellent extensions of rings.

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).

Another Gorenstein Analogue of Projective Complexes

As we know,a complex P is projective if and only if P is exact with Z_n(P)projective in R-Mod for each n∈Z and any morphism f:P→C is null homotopic for any complex C.In this article,we study the notion of DG-Gorenstein projective complexes.We show that a complex G is DG-Gorenstein projective if and only if G is exact with Z_n(G)Gorenstein projective in R-Mod for each n∈Z and any morphism f:G→Q is null homotopic whenever Q is a DG-projective complex.

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.

On Ding Projective Complexes

In the paper, Ding projective modules and Ding projective complexes are considered. In particular, it is proven that Ding projective complexes are precisely the complexes X for which each Xm is a Ding projective R-module for all m ∈ Z.

Some Criteria for the Cohen-Macaulay Property and Local Cohomology

Let a be an ideal of a commutative Noetherian ring R and M be a finitely generated R-module of dimension d. We characterize Cohen-Macaulay rings in term of a special homological dimension. Lastly, we prove that if R is a complete local ring, then the Matlis dual of top local cohomology module Ha^d（M） is a Cohen-Macaulay R-module provided that the R-module M satisfies some conditions.

Some Results on Noetherian Warfield Domains

Let R be a domain.In this paper,we show that if R is one dimensional,then R is a Noetherian Warfield domain if and only if every maximal ideal of R is 2-generated and for every maximal ideal M of R,M is divisorial in the ring(M:M).We also prove that a Noetherian domain R is a Noetherian Warfield domain if and only if for every maximal ideal M of R,M^(2) can be generated by two elements.Finally,we give a sufficient condition under which all ideals of R are strongly Gorenstein projective.