Gorenstein flat preenvelopes and weakly Ding injective covers

TL;DR

This paper investigates conditions under which Gorenstein flat modules form a preenveloping class over certain rings, establishing new links between Ding injective modules, weakly Ding injective modules, and Gorenstein flat modules.

Contribution

It proves that GF is preenveloping over Ding-Chen rings when character modules of Ding injectives are Gorenstein flat, and establishes equivalences involving weakly Ding injectives and their closure properties.

Findings

GF is preenveloping over Ding-Chen rings.

Character modules of Ding injectives being Gorenstein flat implies GF is preenveloping.

Closure of weakly Ding injectives under extensions is equivalent to other module class properties.

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