A remark on prime ideals
If M is a torsion-free module over an integral domain, then we show that for each submodule N of M the envelope EM (N ) of N in M is an essential extension of N. In particular, if N is divisible then EM (N ) = N . The last condition says that N is a semiprime submodule of M if N is proper. Let M be...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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University of Extremadura
2020-06-01
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| Series: | Extracta Mathematicae |
| Subjects: | |
| Online Access: | https://publicaciones.unex.es/index.php/EM/article/view/126 |
| _version_ | 1818886051279667200 |
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| author | S.C. Lee R. Varmazyar |
| author_facet | S.C. Lee R. Varmazyar |
| author_sort | S.C. Lee |
| collection | DOAJ |
| description | If M is a torsion-free module over an integral domain, then we show that for each submodule N of M the envelope EM (N ) of N in M is an essential extension of N. In particular, if N is divisible then EM (N ) = N . The last condition says that N is a semiprime submodule of M if N is proper.
Let M be a module over a ring R such that for any ideals a, b of R, (a ∩ b)M = aM ∩ bM . If N is an irreducible and weakly semiprime submodule of M , then we prove that (N :R M ) is a prime ideal of R. As a result, we obtain that if p is an irreducible ideal of a ring R such that a2 ⊆ p (a is an ideal of R) ⇒ a ⊆ p, then p is a prime ideal. |
| first_indexed | 2024-12-19T16:15:11Z |
| format | Article |
| id | doaj.art-09f9ae1be9824a97acd7152d2effac00 |
| institution | Directory Open Access Journal |
| issn | 0213-8743 2605-5686 |
| language | English |
| last_indexed | 2024-12-19T16:15:11Z |
| publishDate | 2020-06-01 |
| publisher | University of Extremadura |
| record_format | Article |
| series | Extracta Mathematicae |
| spelling | doaj.art-09f9ae1be9824a97acd7152d2effac002022-12-21T20:14:38ZengUniversity of ExtremaduraExtracta Mathematicae0213-87432605-56862020-06-01351A remark on prime idealsS.C. Lee0R. Varmazyar1 Department of Mathematics Education and Institute of Pure and Applied Mathematics Jeonbuk National University, Jeonju, Jeonbuk 54896, South KoreaDepartment of Mathematics, Khoy Branch, Islamic Azad University Khoy 58168-44799, IranIf M is a torsion-free module over an integral domain, then we show that for each submodule N of M the envelope EM (N ) of N in M is an essential extension of N. In particular, if N is divisible then EM (N ) = N . The last condition says that N is a semiprime submodule of M if N is proper. Let M be a module over a ring R such that for any ideals a, b of R, (a ∩ b)M = aM ∩ bM . If N is an irreducible and weakly semiprime submodule of M , then we prove that (N :R M ) is a prime ideal of R. As a result, we obtain that if p is an irreducible ideal of a ring R such that a2 ⊆ p (a is an ideal of R) ⇒ a ⊆ p, then p is a prime ideal.https://publicaciones.unex.es/index.php/EM/article/view/126Prime idealgeneralized prime submodulesemiprime submoduleweakly semiprime submodule |
| spellingShingle | S.C. Lee R. Varmazyar A remark on prime ideals Extracta Mathematicae Prime ideal generalized prime submodule semiprime submodule weakly semiprime submodule |
| title | A remark on prime ideals |
| title_full | A remark on prime ideals |
| title_fullStr | A remark on prime ideals |
| title_full_unstemmed | A remark on prime ideals |
| title_short | A remark on prime ideals |
| title_sort | remark on prime ideals |
| topic | Prime ideal generalized prime submodule semiprime submodule weakly semiprime submodule |
| url | https://publicaciones.unex.es/index.php/EM/article/view/126 |
| work_keys_str_mv | AT sclee aremarkonprimeideals AT rvarmazyar aremarkonprimeideals AT sclee remarkonprimeideals AT rvarmazyar remarkonprimeideals |