On Pseudo-valuation rings and their extensions

• Datum: 2018
• 1. Verfasser: Bhat, V. K.
• Format: Artikel
• Sprache: English
• Veröffentlicht: Lugansk National Taras Shevchenko University 2018
• Schlagworte: Automorphism; derivation; strongly prime ideal; divided prime ideal; pseudo-valuation ring; 16S36; 16N40; 16P40; 16S32
• Online Zugang: https://admjournal.luguniv.edu.ua/index.php/adm/article/view/678
• Назва журналу: Algebra and Discrete Mathematics

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.