NOTES ON STABILITY OF GORENSTEIN CATEGORIES

2012 ◽  
Vol 12 (01) ◽  
pp. 1250209 ◽  
Author(s):  
AIMIN XU
Keyword(s):  
Exact Sequence ◽  
Projective Modules ◽  
Gorenstein Projective Modules ◽  
Gorenstein Categories

Let [Formula: see text] be a class of left R-modules that contains all projective left R-modules. We study a kind of stability of [Formula: see text]-Gorenstein projective modules and settle in affirmative the following question: given an exact sequence of [Formula: see text]-Gorenstein projective left R-modules G = ⋯ → G1 → G0 → G0 → G1 → ⋯ such that the complex Hom R(G, F) is exact for each left R-module [Formula: see text], is the module [Formula: see text]-Gorenstein projective? Some applications are also given.

Stability of Gorenstein Classes of Modules

2013 ◽  
Vol 20 (04) ◽  
pp. 623-636 ◽  
Author(s):  
Samir Bouchiba
Keyword(s):  
Exact Sequence ◽  
Projective Modules ◽  
Natural Question ◽  
Alternate Proof ◽  
Similar Treatment ◽  
Arbitrary Ring ◽  
Injective Modules ◽  
Gorenstein Projective Modules ◽  
Category Of Modules ◽  
Gorenstein Injective

The purpose of this paper is to give, via totally different techniques, an alternate proof to the main theorem of [18] in the category of modules over an arbitrary ring R. In effect, we prove that this theorem follows from establishing a sequence of equalities between specific classes of R-modules. Actually, we tackle the following natural question: What notion emerges when iterating the very process applied to build the Gorenstein projective and Gorenstein injective modules from complete resolutions? In other words, given an exact sequence of Gorenstein injective R-modules G= ⋯ → G1→ G0→ G-1→ ⋯ such that the complex Hom R(H,G) is exact for each Gorenstein injective R-module H, is the module Im (G0→ G-1) Gorenstein injective? We settle such a question in the affirmative and the dual result for the Gorenstein projective modules follows easily via a similar treatment to that used in this paper. As an application, we provide the Gorenstein versions of the change of rings theorems for injective modules over an arbitrary ring.

Ding w-Flat Modules and Dimensions

2018 ◽  
Vol 25 (02) ◽  
pp. 203-216
Author(s):  
Fuad Ali Ahmed Almahdi ◽  
Mohammed Tamekkante
Keyword(s):  
Exact Sequence ◽  
Homological Algebra ◽  
Projective Modules ◽  
Flat Module ◽  
Flat Dimension ◽  
Flat Modules ◽  
Gorenstein Projective Modules ◽  
Gorenstein Homological Algebra

The introduction of w-operation in the class of flat modules has been successful. Let R be a ring. An R-module M is called a w-flat module if [Formula: see text] is GV-torsion for all R-modules N. In this paper, we introduce the w-operation in Gorenstein homological algebra. An R-module M is called Ding w-flat if there exists an exact sequence of projective R-modules … → P1 → P0 → P0 → P1 → … such that M ≅ Im(P0 → P0) and such that the functor HomR(−, F) leaves the sequence exact whenever F is w-flat. Several wellknown classes of rings are characterized in terms of Ding w-flat modules. Some examples are given to show that Ding w-flat modules lie strictly between projective modules and Gorenstein projective modules. The Ding w-flat dimension (of modules and rings) and the existence of Ding w-flat precovers are also studied.

scholarly journals STABILITY OF GORENSTEIN FLAT MODULES

2011 ◽  
Vol 54 (1) ◽  
pp. 169-175 ◽  
Author(s):  
SAMIR BOUCHIBA ◽  
MOSTAFA KHALOUI
Keyword(s):  
Projective Modules ◽  
Natural Question ◽  
Flat Module ◽  
Flat Modules ◽  
Closed Ring ◽  
Gorenstein Projective Modules ◽  
Gorenstein Categories ◽  
Gorenstein Flat ◽  
Gorenstein Injective ◽  
Gorenstein Flat Module

AbstractSather-Wagstaff et al. proved in [8] (S. Sather-Wagsta, T. Sharif and D. White, Stability of Gorenstein categories, J. Lond. Math. Soc.(2), 77(2) (2008), 481–502) that iterating the process used to define Gorenstein projective modules exactly leads to the Gorenstein projective modules. Also, they established in [9] (S. Sather-Wagsta, T. Sharif and D. White, AB-contexts and stability for Goren-stein at modules with respect to semi-dualizing modules, Algebra Represent. Theory14(3) (2011), 403–428) a stability of the subcategory of Gorenstein flat modules under a procedure to build R-modules from complete resolutions. In this paper we are concerned with another kind of stability of the class of Gorenstein flat modules via-à-vis the very Gorenstein process used to define Gorenstein flat modules. We settle in affirmative the following natural question in the setting of a left GF-closed ring R: Given an exact sequence of Gorenstein flat R-modules G = ⋅⋅⋅ G2G1G0G−1G−2 ⋅⋅⋅ such that the complex H ⊗RG is exact for each Gorenstein injective right R-module H, is the module M:= Im(G0 → G−1) a Gorenstein flat module?

Noncommutative G-semihereditary rings

2018 ◽  
Vol 17 (01) ◽  
pp. 1850014 ◽  
Author(s):  
Jian Wang ◽  
Yunxia Li ◽  
Jiangsheng Hu
Keyword(s):  
Associative Ring ◽  
Sufficient Condition ◽  
Projective Modules ◽  
The Third ◽  
Flat Modules ◽  
Gorenstein Projective Modules ◽  
Direct Limits ◽  
Gorenstein Flat ◽  
Finitely Presented ◽  
Semihereditary Rings

In this paper, we introduce and study left (right) [Formula: see text]-semihereditary rings over any associative ring, and these rings are exactly [Formula: see text]-semihereditary rings defined by Mahdou and Tamekkante provided that [Formula: see text] is a commutative ring. Some new characterizations of left [Formula: see text]-semihereditary rings are given. Applications go in three directions. The first is to give a sufficient condition when a finitely presented right [Formula: see text]-module is Gorenstein flat if and only if it is Gorenstein projective provided that [Formula: see text] is left coherent. The second is to investigate the relationships between Gorenstein flat modules and direct limits of finitely presented Gorenstein projective modules. The third is to obtain some new characterizations of semihereditary rings, [Formula: see text]-[Formula: see text] rings and [Formula: see text] rings.

scholarly journals Gorenstein Projective Modules over Triangular Matrix Rings

2016 ◽  
Vol 23 (01) ◽  
pp. 97-104 ◽  
Author(s):  
H. Eshraghi ◽  
R. Hafezi ◽  
Sh. Salarian ◽  
Z. W. Li
Keyword(s):  
Triangular Matrix ◽  
Projective Modules ◽  
Matrix Rings ◽  
Matrix Algebras ◽  
Gorenstein Projective Modules ◽  
Artin Algebras ◽  
Acyclic Complexes ◽  
Matrix Extension

Let R and S be Artin algebras and Γ be their triangular matrix extension via a bimodule SMR. We study totally acyclic complexes of projective Γ-modules and obtain a complete description of Gorenstein projective Γ-modules. We then construct some examples of Cohen-Macaulay finite and virtually Gorenstein triangular matrix algebras.

scholarly journals Gorenstein projective modules and recollements over triangular matrix rings

2020 ◽  
Vol 48 (11) ◽  
pp. 4932-4947 ◽  
Author(s):  
Huanhuan Li ◽  
Yuefei Zheng ◽  
Jiangsheng Hu ◽  
Haiyan Zhu
Keyword(s):  
Triangular Matrix ◽  
Projective Modules ◽  
Matrix Rings ◽  
Gorenstein Projective Modules

scholarly journals The Gorenstein-projective modules over a monomial algebra

2018 ◽  
Vol 148 (6) ◽  
pp. 1115-1134 ◽  
Author(s):  
Xiao-Wu Chen ◽  
Dawei Shen ◽  
Guodong Zhou
Keyword(s):  
Projective Modules ◽  
Stable Category ◽  
Monomial Algebra ◽  
Gorenstein Projective Modules ◽  
The Given

We introduce the notion of a perfect path for a monomial algebra. We classify indecomposable non-projective Gorenstein-projective modules over the given monomial algebra via perfect paths. We apply the classification to a quadratic monomial algebra and describe explicitly the stable category of its Gorenstein-projective modules.

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