Ore Extensions over Weak (Sigma)-rigid Rings and (sigma(*))-rings

TL;DRAbstract

Let $R$ be a ring and $\sigma$ an endomorphism of a ring $R$. Recall that $R$ is said to be a $\sigma(*)$-ring if $a\sigma(a)\in P(R)$ implies $a\in P(R)$ for $a\in R$, where $P(R)$ is the prime radical of $R$. We also recall that $R$ is said to be a weak $\sigma$-rigid ring if $a\sigma(a)\in N(R)$ if and only if $a\in N(R)$ for $a\in R$, where $N(R)$ is the set of nilpotent elements of $R$. In this paper we give a relation between a $\sigma(*)$-ring and a weak $\sigma$-rigid ring. We also give a necessary and sufficient condition for a Noetherian ring to be a weak $\sigma$-rigid ring. Let $\sigma$ be an endomorphism of a ring $R$ and $\delta$ a $\sigma$-derivation of $R$ such that $\sigma(\delta(a)) = \delta(\sigma(a))$ for all $a\in R$. Then $\sigma$ can be extended to an endomorphism (say $\overline{\sigma}$) of $R[x;\sigma,\delta]$ and $\delta$ can be extended to a $\overline{\sigma}$-derivation (say $\overline{\delta}$) of $R[x;\sigma,\delta]$. With this we show that if $R$ is a 2

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