Central Quasi-Morphicity, Central Morphicity, and Strongly $π$-Regularity
Theophilus Gera, Amit Sharma
TL;DR
The paper clarifies when centrally morphic and centrally quasi-morphic modules coincide and how this behavior transfers to the endomorphism ring $S= ext{End}_R(M)$. It shows that if $M$ is image-projective and generates its kernels, then $M$ is centrally morphic, $M$ is centrally quasi-morphic, and $S$ is right centrally morphic are equivalent; for projective $M$, the equivalence reduces to $M$ centrally morphic iff $M$ centrally quasi-morphic and $S$ right centrally morphic. It further proves that a semiprime right centrally quasi-morphic ring with center $Z(R)$ von Neumann regular is strongly $pi$-regular, and provides a module-theoretic analogue: if $Z(S)$ is von Neumann regular and the kernels and images of powers of endomorphisms are fully invariant, then $M$ is strongly $pi$-endoregular iff $S$ is semiprime and $M$ is centrally quasi-morphic. Together, these results correct prior mistaken equivalences and establish a unified framework connecting central (co)morphicity, endomorphism-ring structure, and strong $pi$-regularity.
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 \(π\)-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 \(π\)-endoregular if and only if its endomorphism ring \(S\) is semiprime and \(M\) is centrally quasi-morphic.