n-X-Coherent Rings

TL;DR

This paper introduces a unified concept of n-X-coherent rings that generalizes various existing notions of coherent rings, providing a broad framework and extending classical characterizations to this new setting.

Contribution

It defines n-X-coherent rings for any class of modules and positive integer n, unifying multiple generalizations of coherent rings and extending classical characterizations.

Findings

Unified framework for generalized coherent rings

Extension of classical characterizations to n-X-coherent rings

Applicable to various classes of modules

Abstract

This paper unifies several generalizations of coherent rings in one notion. Namely, we introduce $n$-$\mathscr{X}$-coherent rings, where $\mathscr{X}$ is a class of modules and $n$ is a positive integer, as those rings for which the subclass $\mathscr{X}_n$ of $n$-presented modules of $\mathscr{X}$ is not empty, and every module in $\mathscr{X}_n$ is $n+1$-presented. Then, for each particular class $\mathscr{X}$ of modules, we find correspondent relative coherent rings. Our main aim is to show that the well-known Chase's, Cheatham and Stone's, Enochs', and Stenstrom's characterizations of coherent rings hold true for any $n$-$\mathscr{X}$-coherent rings.