Associated prime ideals of weak σ-rigid 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 δ a σ-derivation of R. Let further σ be such that aσ(a)∈N(R) implies that a∈N(R) for a∈R, where N(R) is the set of nilpotent elements 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 δ(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.