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) ∈ P(R) implies a ∈ P(R) for a ∈ 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) ∈ N(R) if and only if a ∈ N(R) for a ∈ 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 ∈ 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 $Bbb{Q}$ (where $Bbb{Q}$ 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 ∈ R, then R is a weak $sigma$-rigid ring implies that R[x;$sigma,delta$] is a weak $overline{sigma}$-rigid ring. 2000 Mathematics Subject Classifications: 16S36, 16P40, 16P50, 16U20,16W25