The existence of Gorenstein flat covers

Enochs, Edgar E.; Jenda, Overtoun M. G.; López-Ramos, J. A.

doi:10.7146/math.scand.a-14429

Cited by 76 publications

(49 citation statements)

References 15 publications

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“…□ ( It was shown in [15] that every module X has a Gorenstein flat cover over right coherent rings. The existence of Gorenstein flat covers was also proved for modules over more general rings (see [28]).…”

Section: Proposition 319 Let R Be a Ding-chen Ring Andmentioning

confidence: 99%

Homological Properties of Modules Over Ding-Chen Rings

Yang¹

2012

Journal of the Korean Mathematical Society

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Abstract. The so-called Ding-Chen ring is an n-FC ring which is both left and right coherent, and has both left and right self FP-injective dimensions at most n for some non-negative integer n. In this paper, we investigate the classes of the so-called Ding projective, Ding injective and Gorenstein flat modules and show that some homological properties of modules over Gorenstein rings can be generalized to the modules over Ding-Chen rings. We first consider Gorenstein flat and Ding injective dimensions of modules together with Ding injective precovers. We then discuss balance of functors Hom and tensor.

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Section: Proposition 319 Let R Be a Ding-chen Ring Andmentioning

confidence: 99%

Homological Properties of Modules Over Ding-Chen Rings

Yang¹

2012

Journal of the Korean Mathematical Society

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“…For any Gorenstein injective left R-module M , we have M + is a right R-module. Then there exists a Gorenstein flat right R-module F such that F → M + → 0 is exact by [12] since R is left coherent. This gives rise to the exactness of 0 → M ++ → F + .…”

Section: It Suffices To Show That Ifmentioning

confidence: 99%

On Gi-Flat Modules and Dimensions

Gao¹

2013

Journal of the Korean Mathematical Society

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Abstract. Let R be a ring. A right R-module M is called GI-flat if Tor R 1 (M, G) = 0 for every Gorenstein injective left R-module G. It is shown that GI-flat modules lie strictly between flat modules and copure flat modules. Suppose R is an n-FC ring, we prove that a finitely presented right R-module M is GI-flat if and only if M is a cokernel of a Gorenstein flat preenvelope K → F of a right R-module K with F flat. Then we study GI-flat dimensions of modules and rings. Various results in [6] are developed, some new characterizations of von Neumann regular rings are given.

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“…We know from [EJL,Theorem 2.12] that every module in Mod R has a Gorenstein flat cover. For a module M in Mod R, we denote the Gorenstein flat cover of M by GF 0 (M ).…”

Section: Injective Envelopes Of (Gorenstein) Flat Modulesmentioning

confidence: 99%

Injective Envelopes and (Gorenstein) Flat Covers

Enochs

Huang

2011

Algebr Represent Theor

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We characterize left Noetherian rings in terms of the duality property of injective preenvelopes and flat precovers. For a left and right Noetherian ring R, we prove that the flat dimension of the injective envelope of any (Gorenstein) flat left R-module is at most the flat dimension of the injective envelope of R R. Then we get that the injective envelope of R R is (Gorenstein) flat if and only if the injective envelope of every Gorenstein flat left R-module is (Gorenstein) flat, if and only if the injective envelope of every flat left R-module is (Gorenstein) flat, if and only if the (Gorenstein) flat cover of every injective left R-module is injective, and if and only if the opposite version of one of these conditions is satisfied.

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The existence of Gorenstein flat covers

Abstract: We prove that all left modules over a right coherent ring have Gorenstein flat covers.

Cited by 76 publications

References 15 publications

Homological Properties of Modules Over Ding-Chen Rings

Homological Properties of Modules Over Ding-Chen Rings

On Gi-Flat Modules and Dimensions

Injective Envelopes and (Gorenstein) Flat Covers

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