Rings over which the class of Gorenstein flat modules is closed under extensions

TL;DR

This paper studies the properties of Gorenstein flat modules over left GF-closed rings, generalizing known results from right coherent rings and providing methods to construct new examples of such rings.

Contribution

It extends the theory of Gorenstein flat modules to left GF-closed rings, broadening the class of rings where these modules have well-understood properties.

Findings

Gorenstein flat class is projectively resolving over left GF-closed rings.

Characterization of Gorenstein flat modules over left GF-closed rings.

Construction of non-coherent, non-finite weak dimension rings using direct products.

Abstract

A ring $R$ is called left GF-closed, if the class of all Gorenstein flat left $R$-modules is closed under extensions. The class of left GF-closed rings includes strictly the one of right coherent rings and the one of rings of finite weak dimension. In this paper, we investigate the Gorenstein flat dimension over left GF-closed rings. Namely, we generalize the fact that the class of all Gorenstein flat left modules is projectively resolving over right coherent rings to left GF-closed rings. Also, we generalize the characterization of Gorenstein flat left modules (then of Gorenstein flat dimension of left modules) over right coherent rings to left GF-closed rings. Finally, using direct products of rings, we show how to construct a left GF-closed ring that is neither right coherent nor of finite weak dimension.