Notes on Generalizations of Hopfian and Co-Hopfian Modules

Abstract

A module M is called semi co-Hopfian (resp. semi Hopfian) if any injective (resp. surjective) endomorphism of M has a direct summand image (resp. kernel). We show that if M is semi Hopfian strongly co-Hopfian or semi co-Hopfian strongly Hopfian module, then End_R(M) is strongly π-regular ring. As a consequence we obtain a version of Hopkins-Levitzki Theorem extend to semi Hopfian module and to semi co-Hopfian module. The semi Hopficity and semi co-Hopficity of modules over truncated polynomial rings are considered.

Key words and phrases. Semi Hopfian modules, Semi co-Hopfian modules, Dedekind finite modules.

2010 Mathematics Subject Classification. 16P40, 16D10, 16D40