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 $\sigma(*)$-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-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
$\overline{\sigma}$-rigid ring.