Ore Extensions over (σ,δ)-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 $R[x;\sigma ,\delta ]$