On Pseudo-valuation rings and their extensions
Let \(R\) be a commutative Noetherian \(\mathbb{Q}\)-algebra (\(\mathbb{Q}\)is the field of rational numbers). Let \(\sigma\) be an automorphism of \(R\) and \(\delta\) a \(\sigma\)-derivation of \(R\). We define a \(\delta\)-divided ring and prove the following: \((1)\)If \(R\) is a pseudo-valuati...
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| Дата: | 2018 |
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| Формат: | Стаття |
| Мова: | English |
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Lugansk National Taras Shevchenko University
2018
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| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/678 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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admjournalluguniveduua-article-6782018-04-04T09:31:27Z On Pseudo-valuation rings and their extensions Bhat, V. K. Automorphism, derivation, strongly prime ideal, divided prime ideal, pseudo-valuation ring 16S36, 16N40, 16P40, 16S32 Let \(R\) be a commutative Noetherian \(\mathbb{Q}\)-algebra (\(\mathbb{Q}\)is the field of rational numbers). Let \(\sigma\) be an automorphism of \(R\) and \(\delta\) a \(\sigma\)-derivation of \(R\). We define a \(\delta\)-divided ring and prove the following: \((1)\)If \(R\) is a pseudo-valuation ring such that \(x\notin P\) for any prime ideal \(P\) of \(R[x;\sigma,\delta]\), and \(P\cap R\) is a prime ideal of \(R\) with \(\sigma(P\cap R) = P\cap R\) and \(\delta(P\cap R) \subseteq P\cap R\), then \(R[x;\sigma,\delta]\) is also a pseudo-valuation ring. \((2)\)If \(R\) is a \(\delta\)-divided ring such that \(x\notin P\) for any prime ideal \(P\) of \(R[x;\sigma,\delta]\), and \(P\cap R\) is a prime ideal of \(R\) with \(\sigma(P\cap R) = P\cap R\) and \(\delta(P\cap R) \subseteq P\cap R\), then \(R[x;\sigma,\delta]\) is also a \(\delta\)-divided ring. 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/678 Algebra and Discrete Mathematics; Vol 12, No 2 (2011) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/678/212 Copyright (c) 2018 Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics |
| baseUrl_str |
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| datestamp_date |
2018-04-04T09:31:27Z |
| collection |
OJS |
| language |
English |
| topic |
Automorphism derivation strongly prime ideal divided prime ideal pseudo-valuation ring 16S36 16N40 16P40 16S32 |
| spellingShingle |
Automorphism derivation strongly prime ideal divided prime ideal pseudo-valuation ring 16S36 16N40 16P40 16S32 Bhat, V. K. On Pseudo-valuation rings and their extensions |
| topic_facet |
Automorphism derivation strongly prime ideal divided prime ideal pseudo-valuation ring 16S36 16N40 16P40 16S32 |
| format |
Article |
| author |
Bhat, V. K. |
| author_facet |
Bhat, V. K. |
| author_sort |
Bhat, V. K. |
| title |
On Pseudo-valuation rings and their extensions |
| title_short |
On Pseudo-valuation rings and their extensions |
| title_full |
On Pseudo-valuation rings and their extensions |
| title_fullStr |
On Pseudo-valuation rings and their extensions |
| title_full_unstemmed |
On Pseudo-valuation rings and their extensions |
| title_sort |
on pseudo-valuation rings and their extensions |
| description |
Let \(R\) be a commutative Noetherian \(\mathbb{Q}\)-algebra (\(\mathbb{Q}\)is the field of rational numbers). Let \(\sigma\) be an automorphism of \(R\) and \(\delta\) a \(\sigma\)-derivation of \(R\). We define a \(\delta\)-divided ring and prove the following: \((1)\)If \(R\) is a pseudo-valuation ring such that \(x\notin P\) for any prime ideal \(P\) of \(R[x;\sigma,\delta]\), and \(P\cap R\) is a prime ideal of \(R\) with \(\sigma(P\cap R) = P\cap R\) and \(\delta(P\cap R) \subseteq P\cap R\), then \(R[x;\sigma,\delta]\) is also a pseudo-valuation ring. \((2)\)If \(R\) is a \(\delta\)-divided ring such that \(x\notin P\) for any prime ideal \(P\) of \(R[x;\sigma,\delta]\), and \(P\cap R\) is a prime ideal of \(R\) with \(\sigma(P\cap R) = P\cap R\) and \(\delta(P\cap R) \subseteq P\cap R\), then \(R[x;\sigma,\delta]\) is also a \(\delta\)-divided ring. |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2018 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/678 |
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AT bhatvk onpseudovaluationringsandtheirextensions |
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2025-12-02T15:43:08Z |
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2025-12-02T15:43:08Z |
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