### 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, where N(R) is the set of nilpotent elements of R. We recall that a ring R 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.

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