n-X-Coherent Rings

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.

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