Absolutely Self Pure Modules

An $R$-module $M$ is called absolutely self pure if for any finitely generated left ideal of $R$ whose kernel is in the filter generated by the set of all left ideals $L$ of $R$ with $L \supseteq$ ann $(m)$ for some $m \in M$, any map from $L$ to $M$ is a restriction of a map $R \rightarrow M$. Certain properties of quasi injective and absolutely pure modules are extended to absolute self purity. Regular and left noetherian rings are characterized using this new concept.