n-X-Coherent Rings

This paper unifies several generalizations of coherent rings in one notion. Namely, we introduce $n$-$\mathcal{X}$-coherent rings, where $\mathcal{X}$ is a class of modules and $n$ is a positive integer, as those rings for which the subclass ${\mathcal{X}}_n$ of $n$-presented modules of $\mathcal{X}$ is not empty, and every module in ${\mathcal{X}}_n$ is $n+1$-presented. Then, for each particular class $\mathcal{X}$ 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 $n$-$\mathcal{X}$-coherent rings.