Gorenstein flat representations of left rooted quivers

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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Complete Intersection Flat Dimension and the Intersection Theorem

Algebra Colloquium ◽

10.1142/s1005386712000934 ◽

2012 ◽

Vol 19 (spec01) ◽

pp. 1161-1166

Author(s):

Parviz Sahandi ◽

Tirdad Sharif ◽

Siamak Yassemi

Keyword(s):

Local Ring ◽

Complete Intersection ◽

Intersection Theorem ◽

Finitely Generated ◽

Finitely Generated Module ◽

Gorenstein Dimension ◽

Flat Dimension ◽

Gorenstein Flat ◽

Similar Extension

Any finitely generated module M over a local ring R is endowed with a complete intersection dimension CI-dim RM and a Gorenstein dimension G-dim RM. The Gorenstein dimension can be extended to all modules over the ring R. This paper presents a similar extension for the complete intersection dimension, and mentions the relation between this dimension and the Gorenstein flat dimension. In addition, we show that in the intersection theorem, the flat dimension can be replaced by the complete intersection flat dimension.

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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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