S-Prime Right Submodules and an S-Version of Prime Avoidance
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929210073808896 |
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| author | Abouhalaka, Alaa |
| author_facet | Abouhalaka, Alaa |
| contents | Let S be an m-system of a ring R, and P a submodule of a right R-module M. This paper, presents the notion of S-prime submodule and provides some properties and equivalent definitions. We define S-multiplication right module, and prove that in multiplication (S-multiplication) right R-module M, the ideal (P :_R M) is a right S-prime ideal of R if and only if P is an S-prime submodule of M. Moreover, we give an S-version of prime avoidance lemma. Furthermore, we define S-finite and S-Noetherian right modules following the definitions in [1]. We prove that a multiplication finitely generated right R-module M is S-Noetherian if (N :_R M) is an S-prime ideal of R, for all submodules N of M. In addition, we give some examples of right S-Noetherian rings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_07270 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | S-Prime Right Submodules and an S-Version of Prime Avoidance Abouhalaka, Alaa Rings and Algebras 16N60, 16W99, 16D99, 16D25 Let S be an m-system of a ring R, and P a submodule of a right R-module M. This paper, presents the notion of S-prime submodule and provides some properties and equivalent definitions. We define S-multiplication right module, and prove that in multiplication (S-multiplication) right R-module M, the ideal (P :_R M) is a right S-prime ideal of R if and only if P is an S-prime submodule of M. Moreover, we give an S-version of prime avoidance lemma. Furthermore, we define S-finite and S-Noetherian right modules following the definitions in [1]. We prove that a multiplication finitely generated right R-module M is S-Noetherian if (N :_R M) is an S-prime ideal of R, for all submodules N of M. In addition, we give some examples of right S-Noetherian rings. |
| title | S-Prime Right Submodules and an S-Version of Prime Avoidance |
| topic | Rings and Algebras 16N60, 16W99, 16D99, 16D25 |
| url | https://arxiv.org/abs/2401.07270 |