ON (m,n)-PURITY OF MODULES

Abstract

Let R be a ring. Given two positive integers m and n, anR-module V is said to be (m,n)-presented if there is an exact sequence of R-modules 0 → K → Rm → V → 0 withKn-generated. A submodule U of a rightR-module U is said to be (m,n)-pure in U if for every (m,n)presented left R-module V , the canonical map U ⊗R V → U ⊗R V is a monomorphism. A right R-module A is said to be absolutely (m,n)-pure if A is (m,n)-pure in every module which contains A as a submodule. In this paper, several characterizations of (m,n)-purity are given and some properties of (m,n)-purity are investigated, various results of purity are developed, many extending known results. It is shown that a right Rmodule A is absolutely (m,n)-pure if and only if it is (n,m)-injective.