On Ding injective, Ding projective, and Ding flat modules and complexes

TL;DR

This paper characterizes Ding modules and complexes over Ding-Chen rings, showing their equivalence with Gorenstein modules and complexes, and extends known results to broader classes of rings and complexes.

Contribution

It provides a comprehensive characterization of Ding modules and complexes over Ding-Chen rings, linking them to Gorenstein modules and extending existing results to Noetherian and coherent rings.

Findings

Ding projective, injective, and flat modules coincide with Gorenstein modules over Ding-Chen rings.

A complex is Ding projective/injective/flat iff each component is Ding projective/injective/flat.

Exact complexes with Gorenstein injective components have cotorsion cycle modules over Noetherian rings.

Abstract

We characterize Ding modules and complexes over Ding-Chen rings. We show that over a Ding-Chen ring R, the Ding projective (resp. Ding injective, resp. Ding flat) R-modules coincide with the Gorenstein projective (resp. Gorenstein injective, resp. Gorenstein flat) modules, which in turn are nothing more than modules appearing as a cycle of an exact complex of projective (resp. injective, resp. flat) modules. We prove a similar characterization for chain complexes of R-modules: A complex is Ding projective (resp. Ding injective, resp. Ding flat) if and only if each component is Ding projective (resp. Ding injective, resp. Ding flat). Along the way, we generalize some results of Stovicek and Bravo-Gillespie-Hovey to obtain other interesting corollaries. For example, we show that over any Noetherian ring, any exact chain complex with Gorenstein injective components must have all cotorsion cycle modules. That is, Ext(F,ZnI) = 0 for any such complex I and flat module F. On the other hand, over any coherent ring, the cycles of any exact complex P with projective components must satisfy Ext(ZnP,A) = 0 for any absolutely pure module A.