On Pseudo-valuation rings and their extensions

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:

  \((1)\)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.

 \((2)\)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.