Ore Extensions over Weak (Sigma)-rigid Rings and (sigma(*))-rings
Keywords:
Automorphism, (sigma(*))-ring, weak (sigma)-rigid ring, 2-primal ringAbstract
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.
Downloads
Published
Issue
Section
License
Upon acceptance of an article by the journal, the author(s) accept(s) the transfer of copyright of the article to European Journal of Pure and Applied Mathematics.
European Journal of Pure and Applied Mathematics will be Copyright Holder.