Let T be a set of elements in a ring A. The set T is right permutable if for any a ∈ A and t ∈ T, there exist b ∈ A, u ∈ T such that au = tb. A multiplicative set in a ring A is any subset T of A such that 1 ∈ T,0 ∉ T and T is closed under multiplication. A completely prime ideal in a ring A is any proper ideal B such that A\B is a multiplicative set (i.e. A/Bis a domain). A minimal prime ideal (resp. minimal completely prime ideal) in a ring A is any prime (resp. completely prime) ideal P such that P contains no properly any other prime ideal (resp. completely prime ideal) of A. Let I be any proper ideal of a ring A. The set of all elements a ∈ A such that a + I is a regular element of A/I is denoted by c(I). In particular, c(0) is the set of all regular elements of A.

Rings of quotients

In order to research the structure and properties of modulcs overings and algebras,In this paper,by using homological methods we investigated the ralation between Gorenstein injective modules and Gorenstein flat modules,and gave a criteria theorem on Gorenstein flat modules.Meanwhile,we also obtained the con- nection between the Gorenstein flat dimension of the ideal products and modules,and relations among their Gorenstein flat dimension.

On Gorenstein Flat Modules

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Divisor Theory In Module Categories

For a ring R, we prove that all left R-modules have weak injec- tive covers and weak injective preenvelopes. Then we characterize the right weak injective dimension of modules in terms of the properties of left derived functors of Hom and right derived functors of �. For any n � 2, we prove that the global right weak injective dimension of RM is at most n if and only if its global left weak injective dimension is at most n � 2.

Weak injective covers and dimension of modules

Let R be a commutative ring. An R-module M is said to be super finitely presented if there is an exact sequence of R-modules where each Pi is finitely generated projective. In this article, it is shown that if R has the property (B) that every super finitely presented module has finite Gorenstein projective dimension, then every finitely generated Gorenstein projective module is super finitely presented. As an application of the notion of super finitely presented modules, we show that if R has the property (C) that every super finitely presented module has finite projective dimension, then R is K0-regular, i.e., K0(R[x1,…, xn]) ≅ K0(R) for all n ≥ 1.

Super finitely presented modules and Gorenstein projective modules

Let R be a ring. A left R-module M (resp., right R-module N) is called weak injective (resp., weak flat) if (resp., ) for every super finitely presented left R-module F. By replacing finitely presented modules by super finitely presented modules, we may generalize many results of a homological nature from coherent rings to arbitrary rings. Some examples are given to show that weak injective (resp., weak flat) modules need not be FP-injective (resp., not flat) in general. In addition, we introduce and study the super finitely presented dimension (denote by l.sp.gldim(R)) of R that are defined in terms of only super finitely presented left R-modules. Some known results are extended.

Weak Injective and Weak Flat Modules

In this article, Gorenstein FP-injective modules are introduced and investigated. A left R-module M is called Gorenstein FP-injective if there is an exact sequence … → E 1 → E 0 → E 0 → E 1 → … of FP-injective left R-modules with M = ker(E 0 → E 1) such that Hom R (P, −) leaves the sequence exact whenever P is a finitely presented left R-module with pd R (P) < ∞. Some properties of Gorenstein FP-injective modules are obtained. Several well-known classes of rings are characterized in terms of Gorenstein FP-injective modules.

Coherent Rings and Gorenstein FP-Injective Modules

Let R be a ring. A left R-module M is called GI-injective if for any Gorenstein injective left R-module N. It is shown that a left R-module M over any ring R is GI-injective if and only if M is a kernel of a Gorenstein injective precover f: A → B of a left R-module B with A injective. Suppose R is an n-Gorenstein ring, we prove that a left R-module M is GI-injective if and only if M is a direct sum of an injective left R-module and a reduced GI-injective left R-module. Then we investigate GI-injective dimensions of modules and rings. As applications, some new characterizations of the weak (Gorenstein) global dimension of coherent rings are given.

On GI-Injective Modules

This article continues to investigate a particular case of Gorenstein FP-injective modules, called strongly Gorenstein FP-injective modules. Some examples are given to show that strongly Gorenstein FP-injective modules lie strictly between FP-injective modules and Gorenstein FP-injective modules. Various results are developed, many extending known results in [1]. We also characterize FC rings in terms of strongly Gorenstein FP-injective, projective, and flat modules.

On Strongly Gorenstein FP-Injective Modules

Let &#x1d49c; be an abelian category. A subcategory &#x1d4b3; of &#x1d49c; is called coresolving if &#x1d4b3; is closed under extensions and cokernels of monomorphisms and contains all injective objects of &#x1d49c;. In this paper, we introduce and study Gorenstein coresolving categories, which unify the following notions: Gorenstein injective modules [8], Gorenstein FP-injective modules [20], Gorenstein AC-injective modules [3], and so on. Then we define a resolution dimension relative to the Gorenstein coresolving category &#x1d4a2;ℐ&#x1d4b3;(&#x1d49c;). We investigate the properties of the homological dimension and unify some important properties possessed by some known homological dimensions. In addition, we study stability of the Gorenstein coresolving category &#x1d4a2;ℐ&#x1d4b3;(&#x1d49c;) and apply the obtained properties to special subcategories and in particular to module categories.

Zenghui Gao

Papers

Weak Injective and Weak Flat Modules

Coherent Rings and Gorenstein FP-Injective Modules

On GI-Injective Modules

On Strongly Gorenstein FP-Injective Modules

Gorenstein coresolving categories