On Pseudo-valuation rings and their extensions

V. K. Bhat

Algebra and discrete mathematics, Tome 12 (2011) no. 2, pp. 25-30

Résumé

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.

Détail

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