On the nilpotence of the prime radical in module categories
For \(M\in R\)-Mod and \(\tau\) a hereditary torsion theory on the category \(\sigma [M]\) we use the concept of prime and semiprime module defined by Raggi et al. to introduce the concept of \(\tau\)-pure prime radical \(\mathfrak{N}_{\tau}(M) =\mathfrak{N}_{\tau}\) as the intersection of all \(\ta...
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| Date: | 2022 |
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| Main Authors: | , , |
| Format: | Article |
| Language: | English |
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Lugansk National Taras Shevchenko University
2022
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| Online Access: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1634 |
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| Journal Title: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| _version_ | 1856543338669277184 |
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| author | Arellano, C. Castro, J. Ríos, J. |
| author_facet | Arellano, C. Castro, J. Ríos, J. |
| author_sort | Arellano, C. |
| baseUrl_str | |
| collection | OJS |
| datestamp_date | 2022-03-28T05:34:02Z |
| description | For \(M\in R\)-Mod and \(\tau\) a hereditary torsion theory on the category \(\sigma [M]\) we use the concept of prime and semiprime module defined by Raggi et al. to introduce the concept of \(\tau\)-pure prime radical \(\mathfrak{N}_{\tau}(M) =\mathfrak{N}_{\tau}\) as the intersection of all \(\tau\)-pure prime submodules of \(M\). We give necessary and sufficient conditions for the \(\tau\)-nilpotence of \(\mathfrak{N}_{\tau}(M) \). We prove that if \(M\) is a finitely generated \(R\)-module, progenerator in \(\sigma [M]\) and \(\chi\neq \tau\) is FIS-invariant torsion theory such that \(M\) has \(\tau\)-Krull dimension, then \(\mathfrak{N}_{\tau}\) is \(\tau\)-nilpotent. |
| first_indexed | 2026-02-08T08:01:39Z |
| format | Article |
| id | admjournalluguniveduua-article-1634 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2026-02-08T08:01:39Z |
| publishDate | 2022 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| spelling | admjournalluguniveduua-article-16342022-03-28T05:34:02Z On the nilpotence of the prime radical in module categories Arellano, C. Castro, J. Ríos, J. prime modules, semiprime modules, Goldie modules, torsion theory, nilpotent ideal, nilpotence 06F25, 16S90, 16D50, 16P50, 16P70 For \(M\in R\)-Mod and \(\tau\) a hereditary torsion theory on the category \(\sigma [M]\) we use the concept of prime and semiprime module defined by Raggi et al. to introduce the concept of \(\tau\)-pure prime radical \(\mathfrak{N}_{\tau}(M) =\mathfrak{N}_{\tau}\) as the intersection of all \(\tau\)-pure prime submodules of \(M\). We give necessary and sufficient conditions for the \(\tau\)-nilpotence of \(\mathfrak{N}_{\tau}(M) \). We prove that if \(M\) is a finitely generated \(R\)-module, progenerator in \(\sigma [M]\) and \(\chi\neq \tau\) is FIS-invariant torsion theory such that \(M\) has \(\tau\)-Krull dimension, then \(\mathfrak{N}_{\tau}\) is \(\tau\)-nilpotent. Lugansk National Taras Shevchenko University 2022-03-28 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1634 10.12958/adm1634 Algebra and Discrete Mathematics; Vol 32, No 2 (2021) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1634/pdf Copyright (c) 2022 Algebra and Discrete Mathematics |
| spellingShingle | prime modules semiprime modules Goldie modules torsion theory nilpotent ideal nilpotence 06F25 16S90 16D50 16P50 16P70 Arellano, C. Castro, J. Ríos, J. On the nilpotence of the prime radical in module categories |
| title | On the nilpotence of the prime radical in module categories |
| title_full | On the nilpotence of the prime radical in module categories |
| title_fullStr | On the nilpotence of the prime radical in module categories |
| title_full_unstemmed | On the nilpotence of the prime radical in module categories |
| title_short | On the nilpotence of the prime radical in module categories |
| title_sort | on the nilpotence of the prime radical in module categories |
| topic | prime modules semiprime modules Goldie modules torsion theory nilpotent ideal nilpotence 06F25 16S90 16D50 16P50 16P70 |
| topic_facet | prime modules semiprime modules Goldie modules torsion theory nilpotent ideal nilpotence 06F25 16S90 16D50 16P50 16P70 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1634 |
| work_keys_str_mv | AT arellanoc onthenilpotenceoftheprimeradicalinmodulecategories AT castroj onthenilpotenceoftheprimeradicalinmodulecategories AT riosj onthenilpotenceoftheprimeradicalinmodulecategories |