scholarly journals A Note on Gorenstein Flat Dimension

2011 ◽  
Vol 18 (01) ◽  
pp. 155-161 ◽  
Author(s):  
Driss Bennis
Keyword(s):  
Associative Ring ◽  
Projective Dimension ◽  
Flat Dimension ◽  
Coherent Rings ◽  
Gorenstein Flat

Unlike the Gorenstein projective and injective dimensions, the majority of results on the Gorenstein flat dimension have been established only over Noetherian (or coherent) rings. Naturally, one would like to generalize these results to any associative ring. In this direction, we show that the Gorenstein flat dimension is a refinement of the classical flat dimension over any ring; and we investigate the relations between the Gorenstein projective dimension and the Gorenstein flat dimension.

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.

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.

Complete Intersection Flat Dimension and the Intersection Theorem

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.

scholarly journals FLAT RING EPIMORPHISMS OF COUNTABLE TYPE

2019 ◽  
Vol 62 (2) ◽  
pp. 383-439 ◽  
Author(s):  
LEONID POSITSELSKI
Keyword(s):  
Associative Ring ◽  
Abelian Category ◽  
Projective Dimension ◽  
Countable Base ◽  
Linear Topology ◽  
Countable Type ◽  
Flat Ring ◽  
Abelian Categories ◽  
Induced Topology ◽  
Strongly Flat

AbstractLet R→U be an associative ring epimorphism such that U is a flat left R-module. Assume that the related Gabriel topology $\mathbb{G}$ of right ideals in R has a countable base. Then we show that the left R-module U has projective dimension at most 1. Furthermore, the abelian category of left contramodules over the completion of R at $\mathbb{G}$ fully faithfully embeds into the Geigle–Lenzing right perpendicular subcategory to U in the category of left R-modules, and every object of the latter abelian category is an extension of two objects of the former one. We discuss conditions under which the two abelian categories are equivalent. Given a right linear topology on an associative ring R, we consider the induced topology on every left R-module and, for a perfect Gabriel topology $\mathbb{G}$, compare the completion of a module with an appropriate Ext module. Finally, we characterize the U-strongly flat left R-modules by the two conditions of left positive-degree Ext-orthogonality to all left U-modules and all $\mathbb{G}$-separated $\mathbb{G}$-complete left R-modules.

Coherent rings and Gorenstein flat character modules

2020 ◽  
pp. 1-10
Author(s):  
Yanjiong Yang ◽  
Xiaoguang Yan ◽  
Guocheng Dai
Keyword(s):  
Coherent Rings ◽  
Gorenstein Flat

scholarly journals Ascent properties for test modules

2021 ◽  
pp. 153-167
Author(s):  
Keri Sather-Wagstaff
Keyword(s):  
Local Ring ◽  
Projective Dimension ◽  
Graded Algebras ◽  
Ring Homomorphisms ◽  
Flat Dimension ◽  
Finite Dimensional ◽  
Differential Graded Algebras ◽  
Homological Dimensions ◽  
Tor Modules

We investigate modules for which vanishing of Tor-modules implies finiteness of homological dimensions (e.g., projective dimension and G-dimension). In particular, we answer a question of O. Celikbas and Sather-Wagstaff about ascent properties of such modules over residually algebraic flat local ring homomorphisms. To accomplish this, we consider ascent and descent properties over local ring homomorphisms of finite flat dimension, and for flat extensions of finite dimensional differential graded algebras.

K-theory of n-coherent rings

2021 ◽  
pp. 2350007
Author(s):  
Eugenia Ellis ◽  
Rafael Parra
Keyword(s):  
Full Subcategory ◽  
Projective Dimension ◽  
Exact Category ◽  
Finite Projective Dimension ◽  
Coherent Ring ◽  
K Theory ◽  
Coherent Rings

Let [Formula: see text] be a strong [Formula: see text]-coherent ring such that each finitely [Formula: see text]-presented [Formula: see text]-module has finite projective dimension. We consider [Formula: see text] the full subcategory of [Formula: see text]-Mod of finitely [Formula: see text]-presented modules. We prove that [Formula: see text] is an exact category, [Formula: see text] for every [Formula: see text] and we obtain an expression of [Formula: see text].

scholarly journals Cartan-Eilenberg Gorenstein Flat Complexes

2014 ◽  
Vol 114 (1) ◽  
pp. 5 ◽  
Author(s):  
Gang Yang ◽  
Li Liang
Keyword(s):  
Coherent Ring ◽  
Coherent Rings ◽  
Flat Cover ◽  
Gorenstein Flat

In this paper, we study Cartan-Eilenberg Gorenstein flat complexes. We show that over coherent rings a Cartan-Eilenberg Gorenstein flat complex can be gotten by a so-called complete Cartan-Eilenberg flat resolution. We argue that over a coherent ring every complex has a Cartan-Eilenberg Gorenstein flat cover.

A variant theory for the Gorenstein flat dimension

2015 ◽  
Vol 140 (2) ◽  
pp. 183-204
Author(s):  
Samir Bouchiba
Keyword(s):  
Flat Dimension ◽  
Gorenstein Flat
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