# Skew-Laurent Rings over $\sigma(*)$-rings

## Keywords:

Minimal prime, prime radical, automorphism, $\sigma(*)$-ring## Abstract

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

## Published

2014-11-04

## Issue

## Section

Algebra

## License

Upon acceptance of an article by the journal, the author(s) accept(s) the transfer of copyright of the article to *European Journal of Pure and Applied Mathematics.*

*European Journal of Pure and Applied Mathematics will be Copyright Holder.*

## How to Cite

*European Journal of Pure and Applied Mathematics*,

*7*(4), 387-394. https://ejpam.com/index.php/ejpam/article/view/1829