
RESEARCH ARTICLE
Associated prime ideals of weak $\sigma$rigid rings and their extensions
V. K. Bhat^{} ^{} School of Mathematics, SMVD University,
P/o SMVD University, Katra, J and K, India182320
Abstract:
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 ban 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 $P\cap R=U$.
Keywords:
Ore extension, automorphism, derivation, associated prime.
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Bibliographic databases:
MSC: 16XX, 16N40, 16P40, 16S36 Received: 16.10.2009 Revised: 16.10.2009
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Citation:
V. K. Bhat, “Associated prime ideals of weak $\sigma$rigid rings and their extensions”, Algebra Discrete Math., 10:1 (2010), 8–17
Citation in format AMSBIB
\Bibitem{Bha10}
\by V.~K.~Bhat
\paper Associated prime ideals of weak $\sigma$rigid rings and their extensions
\jour Algebra Discrete Math.
\yr 2010
\vol 10
\issue 1
\pages 817
\mathnet{http://mi.mathnet.ru/adm36}
\mathscinet{http://www.ams.org/mathscinetgetitem?mr=2807684}
\zmath{https://zbmath.org/?q=an:1212.16049}
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