Absolutely self pure modules

Abstract

An R-module M is called absolutely self pure if for any finitely generated left ideal L of R, any map from L into M, whose kernel is in the filter generated by the set of all annihilator ideals ann(m) of elements m in M, is a restriction of a map from the ring R into M. Certain properties of quasi injective and absolutely pure modules are extended to absolute self-purity.  Regular, left noetherian, left semihereditary and Pr¨ufer rings are characterized using this concept, for example, a ring R is left noetherian if and only if every absolutely self pure R-module is quasi injective and R is regular if and only if every R-module is absolutely self pure.