FPn-injective and FPn-flat modules with respect to a semidualizing bimodule

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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FG-purity and FG-flat modules

Journal of Algebra and Its Applications ◽

10.1142/s0219498822501675 ◽

2021 ◽

Author(s):

James Gillespie

Keyword(s):

Flat Modules

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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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Finitely Generated Flat Modules and a Characterization of Semiperfect Rings

Communications in Algebra ◽

10.1081/agb-120022787 ◽

2003 ◽

Vol 31 (9) ◽

pp. 4195-4214 ◽

Cited By ~ 10

Author(s):

Alberto Facchini ◽

Dolors Herbera ◽

Iskhak Sakhajev

Keyword(s):

Finitely Generated ◽

Flat Modules

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Flat modules and torsion theories

Mathematische Zeitschrift ◽

10.1007/bf01181621 ◽

1972 ◽

Vol 129 (4) ◽

pp. 331-334 ◽

Cited By ~ 3

Author(s):

William Schelter ◽

Paul Roberts

Keyword(s):

Flat Modules ◽

Torsion Theories

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Flat Modules and Homological Dimensions

Lectures on Modules and Rings - Graduate Texts in Mathematics ◽

10.1007/978-1-4612-0525-8_2 ◽

1999 ◽

pp. 121-205

Author(s):

T. Y. Lam

Keyword(s):

Flat Modules ◽

Homological Dimensions

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Relative FP-gr-injective and gr-flat modules

International Journal of Algebra and Computation ◽

10.1142/s021819671850042x ◽

2018 ◽

Vol 28 (06) ◽

pp. 959-977 ◽

Cited By ~ 1

Author(s):

Tiwei Zhao ◽

Zenghui Gao ◽

Zhaoyong Huang

Keyword(s):

Graded Modules ◽

Flat Modules ◽

Duality Pairs ◽

Coherent Rings ◽

Flat Cover ◽

Finitely Presented

Let [Formula: see text] be an integer. We introduce the notions of [Formula: see text]-FP-gr-injective and [Formula: see text]-gr-flat modules. Then we investigate the properties of these modules by using the properties of special finitely presented graded modules and obtain some equivalent characterizations of [Formula: see text]-gr-coherent rings in terms of [Formula: see text]-FP-gr-injective and [Formula: see text]-gr-flat modules. Moreover, we prove that the pairs (gr-[Formula: see text], gr-[Formula: see text]) and (gr-[Formula: see text], gr-[Formula: see text]) are duality pairs over left [Formula: see text]-coherent rings, where gr-[Formula: see text] and gr-[Formula: see text] denote the subcategories of [Formula: see text]-FP-gr-injective left [Formula: see text]-modules and [Formula: see text]-gr-flat right [Formula: see text]-modules respectively. As applications, we obtain that any graded left (respectively, right) [Formula: see text]-module admits an [Formula: see text]-FP-gr-injective (respectively, [Formula: see text]-gr-flat) cover and preenvelope.

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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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