Skew-Laurent Rings over σ(∗)-rings

Abstract

Let $R$ be an associative ring with identity $1\ne 0$, and $\sigma$  an endomorphism of $R$. We recall $\sigma (\ast )$ property on $R$ (i.e.  $a\sigma (a)\in P(R)$ implies $a\in P(R)$ for $a\in R$, where $P(R)$  is the prime radical of $R$). Also recall that a ring $R$ is said to be  2-primal if and only if $P(R)$ and the set of nilpotent elements of  $R$ coincide, if and only if the prime radical is a completely  semiprime ideal. It can be seen that a $\sigma (\ast )$-ring is a  2-primal ring.   Let $R$ be a ring and $\sigma$ an automorphism of $R$. Then we know that  $\sigma$ can be extended to an automorphism (say $\bar{\sigma }$) of the  skew-Laurent ring $R[x,x^{-1};\sigma ]$. In this paper we show that if $R$ is  a Noetherian ring and $\sigma$ is an automorphism of $R$ such that $R$ is a  $\sigma (\ast )$-ring, then $R[x,x^{-1};\sigma ]$ is a $\bar{\sigma }(\ast )$-ring.  We also prove a similar result for the general Ore extension $R[x;\sigma ,\delta ]$,  where $\sigma$ is an automorphism of $R$ and $\delta$ a $\sigma$-derivation of $R$