Gorenstein Injective and Gorenstein Flat Resolution of Modules Over Gorenstein Rings

Abstract

Let R be a commutative Noetherian ring. There are several characterizations of Gorenstein rings in terms of classical homological dimensions of their modules. In this paper, we use Gorenstein dimensions (Gorenstein injective and Gorenstein flat dimension) to describe Gorenstein rings. Moreover a characterization of Gorenstein injective (resp. Gorenstein flat) modules over Gorenstein rings is given in terms of their Gorenstein flat (resp. Gorenstein injective) resolutions.     

Weak Dimension of FP-injective Modules Over Chain Rings

It is proven that the weak dimension of each FP-injective module over a chain ring which is either Archimedean or not semicoherent is less or equal to 2. This implies that the projective dimension of any countably generated FP-injective module over an Archimedean chain ring is less or equal to 3.     

Gorenstein Modules over Zariski Filtered Rings

Abstract

We study Gorenstein injective and projective modules over Zariski filtered rings and obtain relations between the Gorenstein dimensions on the category of filtered modules from the associated category of graded modules over the associated graded ring.     

Coherent Rings and Gorenstein FP-Injective 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.     

Gorenstein-Projective Modules over Morita Rings

Let △(φ,ψ) =(A BMA ANBB) be a Morita ring which is an Artin algebra.In this paper we investigate the relations between the Gorenstein-projective modules over a Morita ring △(φ,ψ) and the algebras A and B.We prove that if △(φ,ψ) is a Gorenstein algebra and both MA and AN (resp.,both NB and BM) have finite projective dimension,then A (resp.,B) is a Gorenstein algebra.We also discuss when the CM-freeness and the CM-finiteness of a Morita ring △(φ,ψ) is inherited by the algebras A and B.     

Prethick Subcategories of Modules and Characterizations of Local Rings

This article studies characterizing local rings in terms of homological dimensions. The key tool is the notion of a prethick subcategory which we introduce in this article. Our methods recover the theorems of Salarian, Sather-Wagstaff, and Yassemi.     

Gorenstein Injective Envelopes of Artinian Modules

Let R be a commutative Noetherian ring and A an Artinian R-module. We prove that if A has finite Gorenstein injective dimension, then A possesses a Gorenstein injective envelope which is special and Artinian. This, in particular, yields that over a Gorenstein ring any Artinian module possesses a Gorenstein injective envelope which is special and Artinian.     

n- Gorenstein环上的 Gorenstein内射模(英文)

用 Gorenstein内射模刻画了 n-Gorenstein环 .     

Localization of Injective Modules Over Arithmetical Rings

It is proved that localizations of injective R-modules of finite Goldie dimension are injective if R is an arithmetical ring satisfying the following condition: for every maximal ideal P, R _P is either coherent or not semicoherent. If, in addition, each finitely generated R-module has finite Goldie dimension, then localizations of finitely injective R-modules are finitely injective too. Moreover, if R is a Prüfer domain of finite character, localizations of injective R-modules are injective.     

正则环的内射理想

设R为正则环, A∈ M_n(R). 如果M_n(R)AM_n(R)为M_n(R) 的内射理想,该文证明了有U, V∈ GL_n(R)使得UAV为对角矩阵.     

关于正则模的自同态环

本文研究正则模Ｐ的自同态环Ｓ，主要给出使Ｓ的稳定秩为１的Ｐ的几个充分条件．     

环与剩余类环的同调维数

Ｓａｎｄｏｍｉｅｒｓｋｉ Ｆ．Ｌ,Ｓｍａｌｌ Ｌ．Ｗ,和 Ｆｉｅｌｄｓ Ｋ．Ｌ．［１－２］在“幂零”条件下研究了环与约化环的同调维数．然而对一些环（如交换 Ｖｏｎ Ｎｅｕｍａｎｎ正则环）,“幂零’的条件是不成立的．因此,在本文中我们考虑非“幂零”条件下（如Ｒ（Ｒ／Ｉ）（（Ｒ／Ｉ）Ｒ）是Ｒ－投身的或Ｒ（Ｒ／Ｉ）Ｒ是Ｒ－平坦的）,环与约化环的同调维数．     

On the Asymptotic Behavior of the Bass Numbers of Modules

Let (R, 𝔪) be a Noetherian Gorenstein local ring and I be a principal ideal of R. In this article we show that the Bass numbers of the R-modules R/I ⁿ take constant values for large n.     

NOETHERIAN GT-REGULAR RINGS ARE REGULAR

ＮＯＥＴＨＥＲＩＡＮＧＴ－ＲＥＧＵＬＡＲＲＩＮＧＳＡＲＥＲＥＧＵＬＡＲ￥ＬＩＨＵＩＳＨＩ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＳｈａａｎｘｉＮｏｒｍａｌＵｎｉｖｉｓｉｔｙＸｉ＇ａｎ７１００６２，Ｃｈｉｎａ．）Ａｂｓｔｒａｃｔ：Ｉｔｉｓｐｒ...     

Almost-Perfect Rings and Modules

Following [1 Amini , A. , Ershad , M. , Sharif , H. ( 2008 ). Rings over which flat covers of finitely generated modules are projective . Comm. Algebra 36 : 2862 – 2871 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]], a ring R is called right almost-perfect if every flat right R-module is projective relative to R. In this article, we continue the study of these rings and will find some new characterizations of them in terms of decompositions of flat modules. Also we show that a ring R is right almost-perfect if and only if every right ideal of R is a cotorsion module. Furthermore, we prove that over a right almost-perfect ring, every flat module with superfluous radical is projective. Moreover, we define almost-perfect modules and investigate some properties of them.