C-Gorenstein projective, injective and flat modules

Yang, Xiao Yan; Liu, Zhong Kui

doi:10.1007/s10587-010-0077-4

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Theorem 3112

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Cited by 2 publications

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“…By Theorem 3.11, we have that M ∈ GI C if and only if M + ∈ GF C whenever R is an Artinian ring. We note that this has been proven by Yang and Liu in [38,Theorem 3.4]. But it seems that the proof of [38,Theorem 3.4] is incomplete.…”

Section: Theorem 311mentioning

confidence: 73%

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Duality Pairs Induced by Gorenstein Projective Modules with Respect to Semidualizing Modules

Geng

Ding

2015

Algebr Represent Theor

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Let C be a semidualizing module over a commutative Noetherian ring R. We investigate duality pairs induced by C-Gorenstein projective modules. It is proven that R is Artinian if and only if (GP C , GI C ) is a duality pair if and only if (GI C , GP C ) is a duality pair and M + ∈ GI C whenever M ∈ GP C , where GP C (GI C ) is the class of C-Gorenstein projective (C-Gorenstein injective) R-modules. In particular, we give a necessary and sufficient condition for a commutative Artinian ring to be virtually Gorenstein. Moreover, we get that R is Artinian if and only if the class GP of Gorentein projective R-modules is preenveloping. As applications, some new criteria for a semidualizing module to be dualizing are given provided that R is a commutative Artinian ring.

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Section: Theorem 311mentioning

confidence: 73%

“…But it seems that the proof of [38,Theorem 3.4] is incomplete. In the proof of " ⇐ " of [38,Theorem 3.4], the authors successfully construct an exact sequence of the form…”

Section: Theorem 311mentioning

confidence: 99%