Model structures on modules over Ding-Chen rings

TL;DR

This paper introduces Ding-Chen rings, a class of coherent rings with finite self FP-injective dimension, generalizing Gorenstein rings, and develops module theory and model structures for these rings, including group rings.

Contribution

It defines Ding-Chen rings and modules, extending Gorenstein concepts, and generalizes model structures from Gorenstein rings to Ding-Chen rings, including group rings.

Findings

Ding-Chen rings generalize Gorenstein rings.

Standard model structures extend to Ding-Chen rings.

Group rings over Ding-Chen rings are also Ding-Chen.

Abstract

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 in \cite{ding and chen 93} and \cite{ding and chen 96} 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 case the ring is Noetherian 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.