Zhanmin Zhu, Jianlong Chen, Xiaoxiang Zhang


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.

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