Weakly Ding injective preenvelopes and covers

We work over coherent rings. We consider the class of weakly Ding injective modules. These are the cycles of the exact complexes of FP-injective modules that stay exact when applying a functor (Formula presented.) for any FP-injective module A. Throughout the paper, we assume that the class of weakly Ding injective modules, (Formula presented.), is closed under extensions. We prove that any weakly Ding injective module is a direct sum of the form (Formula presented.) with F FP-injective and with D Ding injective. We also prove that the class of weakly Ding injective modules is the right half of a complete hereditary cotorsion pair. Then we show that (Formula presented.) is ccovering if and only if it is closed under direct limits.