A NOTE ON PRIME IDEALS OF IFP-RINGS AND THEIR EXTENSIONS

Abstract

Let R be a ring, σ an automorphism of R and δ a σ-derivation of R. Let further σ be such that aσ(a) ∈ N(R) if and only if a ∈ N(R) for a ∈ R, whereN(R) is the set of nilpotent elements of R. We recall that a ringR is called an IFP-ring if for a, b ∈ R, ab = 0 implies aRb = 0. 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 right Noetherian IFP-ring, which is also an algebra over Q (Q is the field of rational numbers), σ and δ 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 such that (P ∩R)[x;σ,δ]=P and P ∩R = U.