Gorenstein Flat Complexes Over Coherent Rings with Finite Self-FP-Injective Dimension

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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Coherent rings with finite self-FP-injective dimension

Communications in Algebra ◽

10.1080/00927879608825724 ◽

1996 ◽

Vol 24 (9) ◽

pp. 2963-2980 ◽

Cited By ~ 61

Author(s):

Nanqing Ding ◽

Jianlong Chen

Keyword(s):

Injective Dimension ◽

Coherent Rings

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Cartan-Eilenberg Gorenstein Flat Complexes

MATHEMATICA SCANDINAVICA ◽

10.7146/math.scand.a-16637 ◽

2014 ◽

Vol 114 (1) ◽

pp. 5 ◽

Cited By ~ 3

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.

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GORENSTEIN FP-INJECTIVE AND GORENSTEIN FLAT MODULES

Journal of Algebra and Its Applications ◽

10.1142/s0219498808002953 ◽

2008 ◽

Vol 07 (04) ◽

pp. 491-506 ◽

Cited By ~ 39

Author(s):

LIXIN MAO ◽

NANQING DING

Keyword(s):

Exact Sequence ◽

Injective Modules ◽

Flat Modules ◽

Coherent Rings ◽

Gorenstein Flat

In this paper, Gorenstein FP-injective modules are introduced and studied. An R-module M is called Gorenstein FP-injective if there is an exact sequence ⋯ → E1 → E0 → E0 → E1 → ⋯ of injective R-modules with M = ker (E0 → E1) and such that Hom (E, -) leaves the sequence exact whenever E is an FP-injective R-module. Some properties of Gorenstein FP-injective and Gorenstein flat modules over coherent rings are obtained. Several known results are extended.

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A Note on Gorenstein Flat Dimension

Algebra Colloquium ◽

10.1142/s1005386711000095 ◽

2011 ◽

Vol 18 (01) ◽

pp. 155-161 ◽

Cited By ~ 10

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.

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Cohen-Macaulay homological dimensions

MATHEMATICA SCANDINAVICA ◽

10.7146/math.scand.a-119382 ◽

2020 ◽

Vol 126 (2) ◽

pp. 189-208

Author(s):

Parviz Sahandi ◽

Tirdad Sharif ◽

Siamak Yassemi

Keyword(s):

Local Rings ◽

Injective Dimension ◽

Flat Dimension ◽

Homological Dimensions ◽

Gorenstein Injective Dimension ◽

Gorenstein Flat ◽

Gorenstein Injective

We introduce new homological dimensions, namely the Cohen-Macaulay projective, injective and flat dimensions for homologically bounded complexes. Among other things we show that (a) these invariants characterize the Cohen-Macaulay property for local rings, (b) Cohen-Macaulay flat dimension fits between the Gorenstein flat dimension and the large restricted flat dimension, and (c) Cohen-Macaulay injective dimension fits between the Gorenstein injective dimension and the Chouinard invariant.

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