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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2408.16240 |
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| _version_ | 1866911020733169664 |
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| author | Ssevviiri, David Kyomuhangi, Annet |
| author_facet | Ssevviiri, David Kyomuhangi, Annet |
| contents | Let $R$ be a commutative unital ring and $N$ be a submodule of an $R$-module $M$. The submodule $\langle E_M(N)\rangle$ generated by the envelope $E_M(N)$ of $N$ is instrumental in studying rings and modules that satisfy the radical formula. We show that: 1) the semiprime radical is an invariant on all the submodules which are respectively generated by envelopes in the ascending chain of envelopes of a given submodule; 2) for rings that satisfy the radical formula, $\langle E_M(0)\rangle$ is an idempotent radical and it induces a torsion theory whose torsion class consists of all nil $R$-modules and the torsionfree class consists of all reduced $R$-modules; and 3) Noetherian uniserial modules satisfy the semiprime radical formula and their semiprime radical is a nil module. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_16240 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Nil modules and the envelope of a submodule Ssevviiri, David Kyomuhangi, Annet Rings and Algebras Commutative Algebra Algebraic Geometry 16S90, 13C13 Let $R$ be a commutative unital ring and $N$ be a submodule of an $R$-module $M$. The submodule $\langle E_M(N)\rangle$ generated by the envelope $E_M(N)$ of $N$ is instrumental in studying rings and modules that satisfy the radical formula. We show that: 1) the semiprime radical is an invariant on all the submodules which are respectively generated by envelopes in the ascending chain of envelopes of a given submodule; 2) for rings that satisfy the radical formula, $\langle E_M(0)\rangle$ is an idempotent radical and it induces a torsion theory whose torsion class consists of all nil $R$-modules and the torsionfree class consists of all reduced $R$-modules; and 3) Noetherian uniserial modules satisfy the semiprime radical formula and their semiprime radical is a nil module. |
| title | Nil modules and the envelope of a submodule |
| topic | Rings and Algebras Commutative Algebra Algebraic Geometry 16S90, 13C13 |
| url | https://arxiv.org/abs/2408.16240 |