Skew-Laurent Rings over $\sigma(*)$-rings
Keywords:
Minimal prime, prime radical, automorphism, $\sigma(*)$-ringAbstract
Let $R$ be an associative ring with identity $1\neq 0$, and $\sigma$ an endomorphism of $R$. We recall $\sigma(*)$ 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 $\sigma(*)$-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 $\overline{\sigma}$) 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 $\sigma(*)$-ring, then $R[x,x^{-1};\sigma]$ is a $\overline{\sigma}(*)$-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 $R$.Downloads
Published
2014-11-04
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Section
Algebra
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How to Cite
Skew-Laurent Rings over $\sigma(*)$-rings. (2014). European Journal of Pure and Applied Mathematics, 7(4), 387-394. https://www.ejpam.com/index.php/ejpam/article/view/1829