$Associated prime ideals of weak $\sigma$-rigid rings and their extensions$

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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Бібліографічні деталі
Дата:	2018
Автор:	Bhat, V. K.
Формат:	Стаття
Мова:	English
Опубліковано:	Lugansk National Taras Shevchenko University 2018
Теми:	Ore extension automorphism derivation associated prime 16-XX; 16N40 16P40 16S36
Онлайн доступ:	https://admjournal.luguniv.edu.ua/index.php/adm/article/view/638
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Назва журналу:	Algebra and Discrete Mathematics

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Algebra and Discrete Mathematics

id	oai:ojs.admjournal.luguniv.edu.ua:article-638
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spelling	oai:ojs.admjournal.luguniv.edu.ua: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
institution	Algebra and Discrete Mathematics
baseUrl_str
datestamp_date	2018-04-04T09:14:15Z
collection	OJS
language	English
topic	Ore extension automorphism derivation associated prime 16-XX; 16N40 16P40 16S36
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
topic_facet	Ore extension automorphism derivation associated prime 16-XX; 16N40 16P40 16S36
format	Article
author	Bhat, V. K.
author_facet	Bhat, V. K.
author_sort	Bhat, V. K.
title	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_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_sort	associated prime ideals of weak $\sigma$-rigid rings and their extensions
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 $P\cap R = U$.
publisher	Lugansk National Taras Shevchenko University
publishDate	2018
url	https://admjournal.luguniv.edu.ua/index.php/adm/article/view/638
work_keys_str_mv	AT bhatvk associatedprimeidealsofweaksigmarigidringsandtheirextensions
first_indexed	2025-07-17T10:31:24Z
last_indexed	2025-07-17T10:31:24Z
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Associated prime ideals of weak \(\sigma\)-rigid rings and their extensions

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