Ore Extensions over Weak (Sigma)-rigid Rings and (sigma(*))-rings (European Journal of Pure and Applied Mathematics, 2010)
Let $R$ be a ring and $\sigma$ an endomorphism of a ring $R$. Recallthat $R$ 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 radicalof $R$. We also recall that $R$ is said to be a weak $\sigma$-rigidring if $a\sigma(a)\in N(R)$ if and only if $a\in N(R)$ for $a\inR$, where $N(R)$ is the set of nilpotent elements of $R$.In this paper we give a relation between a $\sigma(*)$-ring and aweak $\sigma$-rigid ring. We also give a necessary and sufficientcondition 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 extendedto 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 showthat if $R$ is a 2-primal commutative Noetherian ring which is alsoan algebra over $\mathbb{Q}$ (where $\mathbb{Q}$ is the field ofrational 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.