Abstract
Let R be a ring and m, n two fixed positive integers. In this article, R is defined to be left slightly (m, n)-coherent if each n-generated submodule of the left R-module Rm is (m, n)-projective. It is shown that R is left slightly (m, n)-coherent if and only if the cotorsion pair is hereditary if and only if R is left (m, n)-coherent and the cotorsion pair is hereditary, where (resp., ) stands for the class of all (m, n)-projective (resp., (m, n)-injective) left R-modules, and (resp., denotes the classes of all (m, n)-flat (resp., (m, n)-cotorsion) right R-modules. As applications, (m, n)-homological dimensions of modules and rings over slightly (m, n)-coherent rings are studied.
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