K 0 and rings over which the class of finitely generated strongly gorenstein projective modules is closed under extensions

In this note, we present a survey of results concerning universal deformation rings of finitely generated Gorenstein-projective modules over finite dimensional algebras.

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Quasi-strongly Gorenstein projective modules

Journal of Algebra and Its Applications ◽

10.1142/s0219498819501822 ◽

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.

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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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On finitely generated n-SG-projective modules

Colloquium Mathematicum ◽

10.4064/cm121-1-4 ◽

2010 ◽

Vol 121 (1) ◽

pp. 35-44

Author(s):

Driss Bennis

Keyword(s):

Projective Modules ◽

Finitely Generated

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Gorenstein Projective Modules over Triangular Matrix Rings

Algebra Colloquium ◽

10.1142/s1005386716000122 ◽

2016 ◽

Vol 23 (01) ◽

pp. 97-104 ◽

Cited By ~ 1

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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