(STRONGLY) GORENSTEIN FLAT MODULES OVER GROUP RINGS

AbstractLet $\Gamma $ be a group and ${\Gamma }^{\prime } $ be a subgroup of $\Gamma $ of finite index. Let $M$ be a $\Gamma $-module. It is shown that $M$ is (strongly) Gorenstein flat if and only if it is (strongly) Gorenstein flat as a ${\Gamma }^{\prime } $-module. We also provide some criteria in which the classes of Gorenstein projective and strongly Gorenstein flat $\Gamma $-modules are the same.

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Stability of Strongly Gorenstein Flat Modules

Vietnam Journal of Mathematics ◽

10.1007/s10013-013-0041-3 ◽

2013 ◽

Vol 42 (2) ◽

pp. 171-178 ◽

Cited By ~ 3

Author(s):

Zhanping Wang ◽

Zhongkui Liu

Keyword(s):

Flat Modules ◽

Gorenstein Flat ◽

Strongly Gorenstein Flat Modules

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STRONGLY GORENSTEIN FLAT MODULES

Journal of the Australian Mathematical Society ◽

10.1017/s1446788708000761 ◽

2009 ◽

Vol 86 (3) ◽

pp. 323-338 ◽

Cited By ~ 44

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.

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FPn-Injective, FPn-Flat Covers and Preenvelopes, and Gorenstein AC-Flat Covers

Algebra Colloquium ◽

10.1142/s1005386718000226 ◽

2018 ◽

Vol 25 (02) ◽

pp. 319-334

Author(s):

Daniel Bravo ◽

Sergio Estrada ◽

Alina Iacob

Keyword(s):

Direct Summand ◽

Flat Module ◽

Injective Modules ◽

Flat Modules ◽

Gorenstein Flat ◽

Strongly Gorenstein Flat Modules

We prove that, for any n ≥ 2, the classes of FPn-injective modules and of FPn-flat modules are both covering and preenveloping over any ring R. This includes the case of FP∞-injective and FP∞-flat modules (i.e., absolutely clean and, respectively, level modules). Then we consider a generalization of the class of (strongly) Gorenstein flat modules, i.e., the (strongly) Gorenstein AC-flat modules (cycles of exact complexes of flat modules that remain exact when tensored with any absolutely clean module). We prove that some of the properties of Gorenstein flat modules extend to the class of Gorenstein AC-flat modules; for example, we show that this class is precovering over any ring R. We also show that (as in the case of Gorenstein flat modules) every Gorenstein AC-flat module is a direct summand of a strongly Gorenstein AC-flat module. When R is such that the class of Gorenstein AC-flat modules is closed under extensions, the converse is also true. Moreover, we prove that if the class of Gorenstein AC-flat modules is closed under extensions, then it is covering.

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Strongly Gorenstein flat modules and dimensions

Chinese Annals of Mathematics Series B ◽

10.1007/s11401-011-0659-y ◽

2011 ◽

Vol 32 (4) ◽

pp. 533-548 ◽

Cited By ~ 8

Author(s):

Najib Mahdou ◽

Mohammed Tamekkante

Keyword(s):

Flat Modules ◽

Gorenstein Flat ◽

Strongly Gorenstein Flat Modules

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Noncommutative G-semihereditary rings

Journal of Algebra and Its Applications ◽

10.1142/s0219498818500147 ◽

2018 ◽

Vol 17 (01) ◽

pp. 1850014 ◽

Cited By ~ 2

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.

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Notes on Gorenstein Flat Modules

Bulletin of the Iranian Mathematical Society ◽

10.1007/s41980-018-0135-5 ◽

2018 ◽

Vol 45 (2) ◽

pp. 337-344

Author(s):

Yanjiong Yang ◽

Xiaoguang Yan

Keyword(s):

Flat Modules ◽

Gorenstein Flat

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Gorenstein flat modules relative to injectively resolving subcategories

Journal of Algebra and Its Applications ◽

10.1142/s0219498822500992 ◽

2021 ◽

pp. 2250099

Author(s):

Zenghui Gao ◽

Wan Wu

Keyword(s):

Homological Dimension ◽

Common Generalization ◽

Flat Modules ◽

Resolving Subcategory ◽

Homological Dimensions ◽

Gorenstein Flat

Let [Formula: see text] be an injectively resolving subcategory of left [Formula: see text]-modules. We introduce and study [Formula: see text]-Gorenstein flat modules as a common generalization of some known modules such as Gorenstein flat modules (Enochs, Jenda and Torrecillas, 1993), Gorenstein AC-flat modules (Bravo, Estrada and Iacob, 2018). Then we define a resolution dimension relative to the [Formula: see text]-Gorensteinflat modules, investigate the properties of the homological dimension and unify some important properties possessed by some known homological dimensions. In addition, stability of the category of [Formula: see text]-Gorensteinflat modules is discussed, and some known results are obtained as applications.

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Model structures, recollements and duality pairs

Journal of Algebra and Its Applications ◽

10.1142/s0219498823500172 ◽

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.

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