Ore Extensions over ($\sigma, \delta$)-Rings

Abstract

Let $R$ be a Noetherian, integral domain which is also an algebraover $\mathbb{Q}$ ($\mathbb{Q}$ is the field of rational numbers).Let $\sigma$ be an automorphism of $R$ and $\delta$ a$\sigma$-derivation of $R$. A ring $R$ is called a($\sigma, \delta$)-ring if $a(\sigma(a) + \delta(a)) \in P(R)$implies that $a \in P(R)$ for $a\in R$, where $P(R)$ is the primeradical of $R$. We prove that $R$ is 2-primal if $\delta(P(R))\subseteq P(R)$. We also study the property of minimal prime idealsof $R$ and prove the following in this direction:\\\noindent Let $R$ be a Noetherian, integral domain which is also an algebra over $\mathbb{Q}$. Let $\sigma$ be an automorphism of $R$ and $\delta$ a $\sigma$-derivation of $R$ such that $R$ is a $(\sigma, \delta)$-ring. If $P \in Min.Spec (R)$ is such that$\sigma(P) = P$, then $\delta(P) \subseteq P$. Further if $\delta(P(R)) \subseteq P(R)$, then $P[x; \sigma, \delta]$ is a completely prime ideal of $R[x; \sigma, \delta]$.