Kernel-endoregular modules and the morphic property

Abstract

This paper describes properties of three certain classes of modules M over a ring R determined by conditions on isomorphic direct summands (${\lesssim }^{\oplus }$):

1. The condition that whenever $(Im\lambda \cong )M/\mathit{Ker}\lambda {\lesssim }^{\oplus }M$ then $\mathit{Ker}\lambda$ and $Im\lambda$ are direct summands of M for any endomorphism $\lambda \in \mathit{End}(M)$ (kernel-endoregular modules).

2. The condition that if $M/A\cong B$ where A, $B{\lesssim }^{\oplus }M$ then $M/B\cong A$ (iso-summand-morphic modules).

3. The condition if $M/A\cong B$ where A, $B{\le }^{\oplus }M$, then $M/B\cong A$ (summand-morphic modules) which is precisely the internal cancellation property for modules.

KEYWORDS:

• (co-)Hopfian module
• C2 modules
• (dual-)Rickart module
• D2 modules
• endoregular module
• morphic module
• unit-regular module

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 16D10
• 16D50

Acknowledgments

The authors would like to express their gratitude for the referee’s thorough review of the manuscript and the valuable remarks provided.