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

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 $S={\mathrm{End}}_R(M)$ 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.