Central Quasi-Morphicity, Central Morphicity, and Strongly $\pi$-Regularity

TL;DR

This paper refines the understanding of centrally morphic modules, establishing new equivalences and conditions that connect module properties with the regularity of their endomorphism rings, thus advancing module and ring theory.

Contribution

It corrects and extends previous equivalences between centrally quasi-morphic and morphic modules, linking module properties with endomorphism ring regularity under new conditions.

Findings

Equivalence of being centrally morphic and quasi-morphic under certain conditions

Characterization of strongly π-regular rings via module properties

Conditions for modules to be strongly π-endoregular based on endomorphism ring properties

Abstract

This paper refines the relationship between centrally quasi-morphic and centrally morphic modules, correcting earlier equivalences and extending them to a broader module-theoretic framework. We prove that if a module \(M\) is image-projective and generates its kernels, then the following are equivalent: \(M\) is centrally morphic, \(M\) is centrally quasi-morphic, and its endomorphism ring \(S=\operatorname{End}_R(M)\) is right centrally morphic. This characterization clarifies the role of image-projectivity and kernel-generation in transferring morphic behavior between a module and its endomorphism ring. Furthermore, if \(R\) is a semiprime right centrally quasi-morphic ring with a von Neumann regular center \(Z(R)\), then \(R\) is strongly \(\pi\)-regular. In the module setting, when the endocenter \(Z(S)\) is von Neumann regular and the kernels and images of powers of endomorphisms are fully invariant, an image-projective module \(M\) is strongly \(\pi\)-endoregular if and only if its endomorphism ring \(S\) is semiprime and \(M\) is centrally quasi-morphic.