n-X-COHERENT RINGS

Abstract

This paper unifies several generalizations of coherent rings in one notion. Namely, we introduce n-X -coherent rings, where X is a class of modules and n is a positive integer, as those rings for which the subclass Xn of n-presented modules of X is not empty, and every module in Xn is n + 1-presented. Then, for each particular class 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 Stenstr¨om’s characterizations of coherent rings hold true for any n-X -coherent rings.