Weakly Ding injective modules and complexes

Abstract

We consider two classes of modules over coherent rings: the Gorenstein FP-injective modules, and the weakly Ding injective modules. The Gorenstein FP-injective modules are the cycles of the exact complexes of FP-injective modules. The weakly Ding injective modules are the cycles of the exact complexes of FP-injective modules that stay exact when applying a functor for any FP-injective module A. We prove that the class of Gorenstein FP-injective modules is both covering and preenveloping over any (left) coherent ring with the property that every injective module has finite flat dimension. We also prove that, over the same type of rings, the class of weakly Ding injectives, , is preenveloping in . If, moreover, is closed under extensions, then is a hereditary cotorsion pair. In particular, this is the case when R is a Ding Chen ring, and is closed under extensions. We show that if R is a Ding Chen ring such that every FP-injective module has finite projective dimension, then the two classes of modules coincide. Consequently, the class of weakly Ding injective modules is also covering in this case. In the last part of the paper we prove some analogue results for Gorenstein FP-injective complexes and for weakly Ding injective complexes.