On Weakly Coherent Rings

TL;DR

This paper introduces the concept of weakly coherent rings, explores their properties and transferability under various ring operations, and compares them with strongly 2-coherent rings, providing new examples and insights.

Contribution

It defines weakly coherent rings, investigates their behavior under homomorphic images, extensions, localization, and products, and clarifies their relationship with strongly 2-coherent rings.

Findings

Weakly coherent rings are not necessarily coherent.

The class of weakly coherent rings is not stable under localization.

Weakly coherent rings and strongly 2-coherent rings are not comparable.

Abstract

In this paper, we define weakly coherent rings, and examine the transfer of these rings to homomorphic image, trivial ring extension, localization, and direct product. These results provide examples of weakly coherent rings that are not coherent rings. We show that the class of weakly coherent rings is not stable by localization. Also, we show that the class of weakly coherent rings and the class of strongly 2-coherent rings are not comparable.