Central Quasi-Morphicity, Central Morphicity, and Strongly $π$-Regularity
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866917095299612672 |
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| author | Gera, Theophilus Sharma, Amit |
| author_facet | Gera, Theophilus Sharma, Amit |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_10569 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Central Quasi-Morphicity, Central Morphicity, and Strongly $π$-Regularity Gera, Theophilus Sharma, Amit Rings and Algebras 16D10, 16E50 (Primary) 16S50 (Secondary) 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. |
| title | Central Quasi-Morphicity, Central Morphicity, and Strongly $π$-Regularity |
| topic | Rings and Algebras 16D10, 16E50 (Primary) 16S50 (Secondary) |
| url | https://arxiv.org/abs/2511.10569 |