Skew Polynomial Rings over Weak sigma-rigid Rings and sigma(*)-rings

Abstract

Let R be a ring and σ an endomorphism of R. Recall that R is said to be a σ(∗)-ring if aσ(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 σ-rigid ring if aσ(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 σ(∗)-ring and a weak σ-rigid ring. We also give a necessary and sufficient condition for a Noetherian ring to be a weak σ-rigid ring. Let σ be an endomorphism of a ring R. Then σ can be extended to an endomorphism (say σ) of R[x;σ]. With this we show that if R is a Noetherian ring and σ an automorphism of R, then R is a weak σ-rigid ring if and only if R[x;σ] is a weak σ-rigid ring.