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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
Received: 14.03.2011
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}


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