Gorenstein flat preenvelopes and weakly Ding injective covers

Abstract

We consider a (left) coherent ring R. We prove that if the character module of every Ding injective (left) R-module is Gorenstein flat, then the class of Gorenstein flat (right) R-modules, GF, is preenveloping. We show that this is the case when every injective (left) R-module has finite flat dimension. In particular, GF is preenveloping over any Ding-Chen ring.\\ The proofs use the class of weakly Ding injective (left) R-modules, wDI. We show that, when wDI is closed under extensions, the following statements are equivalent:\\ 1. The character module of every Ding injective left R-module is a Gorenstein flat right R-module.\\ 2. The class of weakly Ding injective left R-modules is closed under direct limits.\\ 3. The class of weakly Ding injective modules is covering.\\ The equivalent statements (1)-(3) imply that GF is preenveloping