# 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 European Journal of Pure and Applied Mathematics, the author(s) retain the copyright to the article. However, by submitting your work, you agree that the article will be published under the Creative Commons Attribution-NonCommercial 4.0 International License (CC BY-NC 4.0). This license allows others to copy, distribute, and adapt your work, provided proper attribution is given to the original author(s) and source. However, the work cannot be used for commercial purposes.

By agreeing to this statement, you acknowledge that:

- You retain full copyright over your work.
- The European Journal of Pure and Applied Mathematics will publish your work under the Creative Commons Attribution-NonCommercial 4.0 International License (CC BY-NC 4.0).
- This license allows others to use and share your work for non-commercial purposes, provided they give appropriate credit to the original author(s) and source.

## How to Cite

*European Journal of Pure and Applied Mathematics*,

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