Associated prime ideals of weak \(\sigma\)-rigid rings and their extensions
Let \(R\) be a right Noetherian ring which is also an algebra over \(\mathbb{Q}\) (\(\mathbb{Q}\) the field of rational numbers). Let \(\sigma\) be an automorphism of R and \(\delta\) a \(\sigma\)-derivation of \(R\). Let further \(\sigma\) be such that \(a\sigma(a)\in N(R)\) implies that...
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| Date: | 2018 |
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| Format: | Article |
| Language: | English |
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Lugansk National Taras Shevchenko University
2018
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| Online Access: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/638 |
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| Journal Title: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| _version_ | 1856543354685227008 |
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| author | Bhat, V. K. |
| author_facet | Bhat, V. K. |
| author_sort | Bhat, V. K. |
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| collection | OJS |
| datestamp_date | 2018-04-04T09:14:15Z |
| description | Let \(R\) be a right Noetherian ring which is also an algebra over \(\mathbb{Q}\) (\(\mathbb{Q}\) the field of rational numbers). Let \(\sigma\) be an automorphism of R and \(\delta\) a \(\sigma\)-derivation of \(R\). Let further \(\sigma\) be such that \(a\sigma(a)\in N(R)\) implies that \(a\in N(R)\) for \(a\in R\), where \(N(R)\) is the set of nilpotent elements of \(R\). In this paper we study the associated prime ideals of Ore extension \(R[x;\sigma,\delta]\) and we prove the following in this direction:Let \(R\) be a semiprime right Noetherian ring which is also an algebra over \(\mathbb{Q}\). Let \(\sigma\) and \(\delta\) be as above. Then \(P\) is an associated prime ideal of \(R[x;\sigma,\delta]\) (viewed as a right module over itself) if and only if there exists an associated prime ideal \(U\) of \(R\) with \(\sigma(U) = U\) and \(\delta(U)\subseteq U\) and \(P = U[x;\sigma,\delta]\).We also prove that if \(R\) be a right Noetherian ring which is also an algebra over \(\mathbb{Q}\), \(\sigma\) and \(\delta\) as usual such that \(\sigma(\delta(a))=\delta(\sigma(a))\) for all \(a\in R\) and \(\sigma(U) = U\) for all associated prime ideals \(U\) of \(R\) (viewed as a right module over itself), then \(P\) is an associated prime ideal of \(R[x;\sigma,\delta]\) (viewed as a right module over itself) if and only if there exists an associated prime ideal \(U\) of \(R\) such that \((P\cap R)[x;\sigma,\delta] = P\) and \(P\cap R = U\). |
| first_indexed | 2026-02-08T07:57:59Z |
| format | Article |
| id | admjournalluguniveduua-article-638 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2026-02-08T07:57:59Z |
| publishDate | 2018 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| spelling | admjournalluguniveduua-article-6382018-04-04T09:14:15Z Associated prime ideals of weak \(\sigma\)-rigid rings and their extensions Bhat, V. K. Ore extension, automorphism, derivation, associated prime 16-XX; 16N40, 16P40, 16S36 Let \(R\) be a right Noetherian ring which is also an algebra over \(\mathbb{Q}\) (\(\mathbb{Q}\) the field of rational numbers). Let \(\sigma\) be an automorphism of R and \(\delta\) a \(\sigma\)-derivation of \(R\). Let further \(\sigma\) be such that \(a\sigma(a)\in N(R)\) implies that \(a\in N(R)\) for \(a\in R\), where \(N(R)\) is the set of nilpotent elements of \(R\). In this paper we study the associated prime ideals of Ore extension \(R[x;\sigma,\delta]\) and we prove the following in this direction:Let \(R\) be a semiprime right Noetherian ring which is also an algebra over \(\mathbb{Q}\). Let \(\sigma\) and \(\delta\) be as above. Then \(P\) is an associated prime ideal of \(R[x;\sigma,\delta]\) (viewed as a right module over itself) if and only if there exists an associated prime ideal \(U\) of \(R\) with \(\sigma(U) = U\) and \(\delta(U)\subseteq U\) and \(P = U[x;\sigma,\delta]\).We also prove that if \(R\) be a right Noetherian ring which is also an algebra over \(\mathbb{Q}\), \(\sigma\) and \(\delta\) as usual such that \(\sigma(\delta(a))=\delta(\sigma(a))\) for all \(a\in R\) and \(\sigma(U) = U\) for all associated prime ideals \(U\) of \(R\) (viewed as a right module over itself), then \(P\) is an associated prime ideal of \(R[x;\sigma,\delta]\) (viewed as a right module over itself) if and only if there exists an associated prime ideal \(U\) of \(R\) such that \((P\cap R)[x;\sigma,\delta] = P\) and \(P\cap R = U\). Lugansk National Taras Shevchenko University 2018-04-04 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/638 Algebra and Discrete Mathematics; Vol 10, No 1 (2010) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/638/172 Copyright (c) 2018 Algebra and Discrete Mathematics |
| spellingShingle | Ore extension automorphism derivation associated prime 16-XX; 16N40 16P40 16S36 Bhat, V. K. Associated prime ideals of weak \(\sigma\)-rigid rings and their extensions |
| title | Associated prime ideals of weak \(\sigma\)-rigid rings and their extensions |
| title_full | Associated prime ideals of weak \(\sigma\)-rigid rings and their extensions |
| title_fullStr | Associated prime ideals of weak \(\sigma\)-rigid rings and their extensions |
| title_full_unstemmed | Associated prime ideals of weak \(\sigma\)-rigid rings and their extensions |
| title_short | Associated prime ideals of weak \(\sigma\)-rigid rings and their extensions |
| title_sort | associated prime ideals of weak \(\sigma\)-rigid rings and their extensions |
| topic | Ore extension automorphism derivation associated prime 16-XX; 16N40 16P40 16S36 |
| topic_facet | Ore extension automorphism derivation associated prime 16-XX; 16N40 16P40 16S36 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/638 |
| work_keys_str_mv | AT bhatvk associatedprimeidealsofweaksigmarigidringsandtheirextensions |