Prime ideals of σ(*)-rings and their extensions

Abstract

Let R be a right Noetherian ring which is also an algebra over ℚ (ℚ the field of rational numbers). Let σ be an automorphism of R and σ a σ-derivation of R. Let further σ be such that aσ(a) ∈ P(R) implies that a ∈ P(R) for a ∈ R, where P(R) is the prime radical of R. In this paper we study the associated prime ideals of Ore extension R[x; σ, δ] and we prove the following in this direction: Let R be a semiprime right Noetherian ring which is also an algebra over ℚ. Let σ and δ be as above. Then P is an associated prime ideal of R[x; σ, δ] (viewed as a right module over itself) if and only if there exists an associated prime ideal U of R with σ(U) = U and σ(U) ⊆ U and P = U[x; σ, δ]. We also prove that if R be a right Noetherian ring which is also an algebra over ℚ, σ and δ as usual such that σ(δ(a)) = δ(σ(a)) for all a ∈ R and σ(U) = U for all associated prime ideals U of R (viewed as a right module over itself), then P is an associated prime ideal of R[x; σ, δ] (viewed as a right module over itself) if and only if there exists an associated prime ideal U of R such that (P ∩ R)[x; σ, δ] = P and P ∩ R = U.