Minimal prime ideals of $\sigma(*)$-rings and their extensions. Armenian Journal of Mathematics

Abstract

Let $R$ be a right Noetherian ring which is also an algebra over $\mathbb{Q}$ ($\mathbb{Q}$ the field of rational numbers). Let $\sigma$ be an automorphism of R and $\delta$ a $\sigma$-derivation of $R$. Let further $\sigma$ be such that $a\sigma(a)\in P(R)$ implies that $a\in P(R)$ for $a\in R$, where $P(R)$ is the prime radical of $R$. In this paper we study minimal prime ideals of Ore extension $R[x;\sigma,\delta]$ and we prove the following in this direction: Let $R$ be a right Noetherian ring which is also an algebra over $\mathbb{Q}$. Let $\sigma$ and $\delta$ be as above. Then $P$ is a minimal prime ideal of $R[x;\sigma,\delta]$ if and only if there exists a minimal prime ideal $U$ of $R$ with $P = U[x;\sigma,\delta]$.