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 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?

Model structures, recollements and duality pairs

2021 ◽  
Author(s):  
Wenjing Chen ◽  
Zhongkui Liu
Keyword(s):  
Projective Modules ◽  
Cotorsion Pair ◽  
Gorenstein Rings ◽  
Injective Modules ◽  
Flat Modules ◽  
Duality Pairs ◽  
Cotorsion Pairs ◽  
Gorenstein Flat ◽  
Model Structures

In this paper, we construct some model structures corresponding Gorenstein [Formula: see text]-modules and relative Gorenstein flat modules associated to duality pairs, Frobenius pairs and cotorsion pairs. By investigating homological properties of Gorenstein [Formula: see text]-modules and some known complete hereditary cotorsion pairs, we describe several types of complexes and obtain some characterizations of Iwanaga–Gorenstein rings. Based on some facts given in this paper, we find new duality pairs and show that [Formula: see text] is covering as well as enveloping and [Formula: see text] is preenveloping under certain conditions, where [Formula: see text] denotes the class of Gorenstein [Formula: see text]-injective modules and [Formula: see text] denotes the class of Gorenstein [Formula: see text]-flat modules. We give some recollements via projective cotorsion pair [Formula: see text] cogenerated by a set, where [Formula: see text] denotes the class of Gorenstein [Formula: see text]-projective modules. Also, many recollements are immediately displayed through setting specific complete duality pairs.

scholarly journals STRONGLY GORENSTEIN FLAT MODULES

2009 ◽  
Vol 86 (3) ◽  
pp. 323-338 ◽  
Author(s):  
NANQING DING ◽  
YUANLIN LI ◽  
LIXIN MAO
Keyword(s):  
Exact Sequence ◽  
Projective Modules ◽  
Flat Dimension ◽  
Flat Modules ◽  
Coherent Rings ◽  
Gorenstein Flat ◽  
Strongly Gorenstein Flat Modules

AbstractIn this paper, strongly Gorenstein flat modules are introduced and investigated. An R-module M is called strongly Gorenstein flat if there is an exact sequence ⋯→P1→P0→P0→P1→⋯ of projective R-modules with M=ker (P0→P1) such that Hom(−,F) leaves the sequence exact whenever F is a flat R-module. Several well-known classes of rings are characterized in terms of strongly Gorenstein flat modules. Some examples are given to show that strongly Gorenstein flat modules over coherent rings lie strictly between projective modules and Gorenstein flat modules. The strongly Gorenstein flat dimension and the existence of strongly Gorenstein flat precovers and pre-envelopes are also studied.

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.

Quasi-strongly Gorenstein projective modules

2019 ◽  
Vol 18 (10) ◽  
pp. 1950182
Author(s):  
Kui Hu ◽  
Fanggui Wang ◽  
Longyu Xu ◽  
Dechuan Zhou
Keyword(s):  
Dedekind Domain ◽  
Projective Modules ◽  
Finitely Generated ◽  
Prüfer Domain ◽  
Dedekind Domains ◽  
Gorenstein Projective Modules ◽  
Semihereditary Rings

In this paper, we introduce the class of quasi-strongly Gorenstein projective modules which is a particular subclass of the class of finitely generated Gorenstein projective modules. We also introduce and characterize quasi-strongly Gorenstein semihereditary rings. We call a quasi-strongly Gorenstein semihereditary domain a quasi-SG-Prüfer domain. A Noetherian quasi-SG-Prüfer domain is called a quasi-strongly Gorenstein Dedekind domain. Let [Formula: see text] be a field and [Formula: see text] be an indeterminate over [Formula: see text]. We prove that every ideal of the ring [Formula: see text] is strongly Gorenstein projective. We also show that every ideal of the ring [Formula: see text] (respectively, [Formula: see text]) is strongly Gorenstein projective. These domains are examples of quasi-strongly Gorenstein Dedekind domains.

scholarly journals Super Finitely Presented Modules and Gorenstein Projective Modules

2016 ◽  
Vol 44 (9) ◽  
pp. 4056-4072 ◽  
Author(s):  
Fanggui Wang ◽  
Lei Qiao ◽  
Hwankoo Kim
Keyword(s):  
Projective Modules ◽  
Gorenstein Projective Modules ◽  
Finitely Presented

Some conditions for the existence of Gorenstein projective covers and preenvelopes

2016 ◽  
Vol 15 (08) ◽  
pp. 1650156
Author(s):  
Bin Yu
Keyword(s):  
Commutative Ring ◽  
Projective Modules ◽  
Gorenstein Projective Modules ◽  
Gorenstein Flat ◽  
Gorenstein Injective ◽  
Projective Covers

In this paper, we investigate the rings over which a module is Gorenstein flat if and only if it is Gorenstein projective. Some examples of such rings are given. We show that over such rings the class of Gorenstein projective modules is covering. We also characterize the rings over which the class of Gorenstein projective modules is preenveloping. As a conclusion, we obtain that a commutative ring is artinian if and only if every module has a Gorenstein projective preenvelope. The existence of pure injective Gorenstein injective preenvelopes over certain rings is also shown.

Notes on Gorenstein Flat Modules

2018 ◽  
Vol 45 (2) ◽  
pp. 337-344
Author(s):  
Yanjiong Yang ◽  
Xiaoguang Yan
Keyword(s):  
Flat Modules ◽  
Gorenstein Flat

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.

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