We characterize Ding modules and complexes over Ding-Chen rings. We show thatover a Ding-Chen ring R, the Ding projective (resp. Ding injective, resp. Dingflat) R-modules coincide with the Gorenstein projective (resp. Gorensteininjective, resp. Gorenstein flat) modules, which in turn are nothing more thanmodules appearing as a cycle of an exact complex of projective (resp.injective, resp. flat) modules. We prove a similar characterization for chaincomplexes of R-modules: A complex is Ding projective (resp. Ding injective,resp. Ding flat) if and only if each component is Ding projective (resp. Dinginjective, resp. Ding flat). Along the way, we generalize some results ofStovicek and Bravo-Gillespie-Hovey to obtain other interesting corollaries. Forexample, we show that over any Noetherian ring, any exact chain complex withGorenstein 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 projectivecomponents must satisfy Ext(ZnP,A) = 0 for any absolutely pure module A.
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