σ(∗)-RINGS AND THEIR EXTENSIONS AS 2-PRIMAL RINGS

Abstract

In this article, we discuss the prime radical of skew polynomial rings over Noetherian rings. We recall σ(∗) property on a ring R (i.e. aσ(a) ∈ P(R) implies a ∈ P(R) for a ∈ R, where P(R) is the prime radical of R), where σ is an endomorphism of R. Also recall that a ring R is 2-primal if and only if P(R) and the set of nilpotent elements of R are same, if and only if the prime radical is a completely semiprime ideal. It can be seen that a σ(∗) is a 2-primal ring. In this article we show that if R is a Noetherian σ(∗)-ring, which is also an algebra over Q; σ is an automorphism of R and δ a σ-derivation of R, then the Ore extension R[x; σ, δ] is 2-primal Noetherian.