Some characterizations of coherent rings in terms of strongly FP-injective modules

Abstract

An R-module M is called strongly FP-injective if for any finitely presented R-module P and any i > 0. Denoted by the class of all strongly FP-injective R-modules and by the left orthogonal class with respect to Comparing with some classical results of Noetherian rings, we show that a ring is coherent if and only if is closed under pure submodules; if and only if any finitely generated module in is finitely presented; if and only if R is -coherent and is closed under direct sums; if and only if for any nonnegative integer n, any finitely presented left R-module X, any R-T-bimodule M and any injective right T-module E. In addition, we show that a module is injective if and only if it is a pure -periodic module, where denotes the class of all injective modules.