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 Algebra Discrete Math.: Year: Volume: Issue: Page: Find

 Algebra Discrete Math., 2011, Volume 12, Issue 2, Pages 25–30 (Mi adm113)

RESEARCH ARTICLE

On Pseudo-valuation rings and their extensions

V. K. Bhat

School of Mathematics, SMVD University, P/o SMVD University, Katra, J and K, India 182320

Abstract: 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:
• 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.
• 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.

Keywords: automorphism, derivation, strongly prime ideal, divided prime ideal, pseudo-valuation ring.

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Bibliographic databases:
MSC: 16S36, 16N40, 16P40, 16S32
Revised: 14.03.2011
Language:

Citation: V. K. Bhat, “On Pseudo-valuation rings and their extensions”, Algebra Discrete Math., 12:2 (2011), 25–30

Citation in format AMSBIB
\Bibitem{Bha11} \by V.~K.~Bhat \paper On Pseudo-valuation rings and their extensions \jour Algebra Discrete Math. \yr 2011 \vol 12 \issue 2 \pages 25--30 \mathnet{http://mi.mathnet.ru/adm113} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2952898} \zmath{https://zbmath.org/?q=an:1257.16018}