scholarly journals A GENERALIZATION OF STRONGLY GORENSTEIN PROJECTIVE MODULES

2009 ◽  
Vol 08 (02) ◽  
pp. 219-227 ◽  
Author(s):  
DRISS BENNIS ◽  
NAJIB MAHDOU
Keyword(s):  
Noetherian Rings ◽  
Projective Modules ◽  
Gorenstein Projective Modules

This paper generalize the idea of the authors in J. Pure Appl. Algebra210 (2007) 437–445. Namely, we define and study a particular case of Gorenstein projective modules. We investigate some change of rings results for this new kind of modules. Examples over not necessarily Noetherian rings are given.

On relative counterpart of Auslander’s conditions

2021 ◽  
Author(s):  
Driss Bennis ◽  
Rachid El Maaouy ◽  
J. R. García Rozas ◽  
Luis Oyonarte
Keyword(s):  
Noetherian Rings ◽  
Projective Modules ◽  
New Approach ◽  
Gorenstein Rings ◽  
Semidualizing Module ◽  
Gorenstein Projective Modules ◽  
Frobenius Category ◽  
Relative Notion ◽  
Frobenius Categories

It is now well known that the conditions used by Auslander to define the Gorenstein projective modules on Noetherian rings are independent. Recently, Ringel and Zhang adopted a new approach in investigating Auslander’s conditions. Instead of looking for examples, they investigated rings on which certain implications between Auslander’s conditions hold. In this paper, we investigate the relative counterpart of Auslander’s conditions. So, we extend Ringel and Zhang’s work and introduce other concepts. Namely, for a semidualizing module [Formula: see text], we introduce weakly [Formula: see text]-Gorenstein and partially [Formula: see text]-Gorenstein rings as rings representing relations between the relative counterpart of Auslander’s conditions. Moreover, we introduce a relative notion of the well-known Frobenius category. We show how useful are [Formula: see text]-Frobenius categories in characterizing weakly [Formula: see text]-Gorenstein and partially [Formula: see text]-Gorenstein rings.

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.

Strict Mittag–Leffler modules over Gorenstein injective modules

2019 ◽  
Vol 19 (03) ◽  
pp. 2050050 ◽  
Author(s):  
Yanjiong Yang ◽  
Xiaoguang Yan
Keyword(s):  
Noetherian Ring ◽  
Direct Limit ◽  
Injective Module ◽  
Noetherian Rings ◽  
Projective Modules ◽  
Injective Modules ◽  
Pure Submodules ◽  
Covering Class ◽  
Gorenstein Injective ◽  
Finitely Presented

In this paper, we study the conditions under which a module is a strict Mittag–Leffler module over the class [Formula: see text] of Gorenstein injective modules. To this aim, we introduce the notion of [Formula: see text]-projective modules and prove that over noetherian rings, if a module can be expressed as the direct limit of finitely presented [Formula: see text]-projective modules, then it is a strict Mittag–Leffler module over [Formula: see text]. As applications, we prove that if [Formula: see text] is a two-sided noetherian ring, then [Formula: see text] is a covering class closed under pure submodules if and only if every injective module is strict Mittag–Leffler over [Formula: see text].

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

3.—A Note on Projective Modules

1976 ◽  
Vol 75 (1) ◽  
pp. 24-31 ◽  
Author(s):  
P. F. Smith
Keyword(s):  
Noetherian Rings ◽  
Projective Modules ◽  
Finitely Generated ◽  
Right Noetherian Rings

SynopsisFor various classes of right noetherian rings it is shown that projective right modules are either finitely generated or free.

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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