### ON (m, n)-PURITY OF MODULES

#### Abstract

Let R be a ring. Given two positive integers m and n,an R-module V

is said to be (m, n)-presented if there is an exact sequence of R-modules

0 ?K?R^m?V ?0 with K n-generated. A submodule U' of

a right R-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 R module A is absolutely (m, n)-pure if and only if it is (n, m)-injective.

is said to be (m, n)-presented if there is an exact sequence of R-modules

0 ?K?R^m?V ?0 with K n-generated. A submodule U' of

a right R-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 R module A is absolutely (m, n)-pure if and only if it is (n, m)-injective.

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