Model structures on modules over Ding-Chen rings

An $n$-FC ring is a left and right coherent ring whose left and right self-FP-injective dimension is n. The work of Ding and Chen shows that these rings possess properties which generalize those of $n$-Gorenstein rings. In this paper we call a (left and right) coherent ring with finite (left and right) self-FP-injective dimension a Ding-Chen ring. In the case of Noetherian rings, these are exactly the Gorenstein rings. We look at classes of modules we call Ding projective, Ding injective and Ding flat which are meant as analogs to Enochs’ Gorenstein projective, Gorenstein injective and Gorenstein flat modules. We develop basic properties of these modules. We then show that each of the standard model structures on Mod-$R$, when $R$ is a Gorenstein ring, generalizes to the Ding-Chen case. We show that when $R$ is a commutative Ding-Chen ring and $G$ is a finite group, the group ring $R[G]$ is a Ding-Chen ring.

Keywords

model structure, Ding-Chen ring

2010 Mathematics Subject Classification

18G35, 55U35

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Published 1 January 2010

This article, as originally published in print, specified an incorrect “Received” date of October 10, 2010. The correct “Received” date is October 10, 2009, which appears in this online version of the article.