SKEW POLYNOMIAL RINGS OVER 2-PRIMAL NOETHERIAN RINGS

Abstract

Let R be a ring and σ an automorphism of R and δ a σ-derivation of R. We say that R is a δ-ring if aδ(a) ∈ P(R) implies a ∈ P(R), where P(R) is the prime radical of R. We prove that R[x;σ,δ] is a 2-primal Noetherian ring if R is a Noetherian ring, which moreover an algebra over the field of rational numbers, σ and δ are such that R is a δ-ring and σ(P)=P, P being any minimal prime ideal of R. We use this to prove that if R is a Noetherian σ(∗)-ring (i.e. aσ(a) ∈ P(R) implies a ∈ P(R)), δ a σ-derivation of R such that R is a δ-ring, then R[x;σ,δ] is a 2-primal Noetherian ring.