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

Abstract

Let $R$ be a ring and $\sigma$ an endomorphism of a ring $R$. Recall
that $R$ is said to be a $\sigma (\ast )$-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 $\sigma (\ast )$-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 $\bar{\sigma }$) of $R[x;\sigma ,\delta ]$ and $\delta$ can be extended to a $\bar{\sigma }$-derivation
(say $\bar{\delta }$) of $R[x;\sigma ,\delta ]$. With this we show
that if $R$ is a 2-primal commutative Noetherian ring which is also
an algebra over $\mathbb{Q}$ (where $\mathbb{Q}$ is the field of
rational numbers), $\sigma$ is an automorphism of $R$ and $\delta$ a
$\sigma$-derivation of $R$ such that $\sigma (\delta (a))=\delta (\sigma (a))$ for all $a\in R$, then $R$ is a weak
$\sigma$-rigid ring implies that $R[x;\sigma ,\delta ]$ is a weak
$\bar{\sigma }$-rigid ring.