ASSOCIATED PRIME IDEALS OF A PRINCIPAL IDEAL

Abstract. Let R be an integral domain with identity. We show that each associated prime ideal of a principal ideal in R[X] has height one if and only if each associated prime ideal of a principal ideal in R has height one and R is an S-domain. Krull’s principal ideal theorem [7, Theorem 142] states that for a nonunit element x of a Noetherian ring R, if P is a prime ideal of R which is minimal over xR, then the height of P is at most one. Thus if R is a Noetherian domain then each minimal prime ideal of a nonzero principal ideal has height one. In [2], Barucci-Anderson-Dobbs studied integral domains in which each prime ideal over a nonzero principal ideal has height one. As [2], we say that an integral domain R satisfies the principal ideal theorem (PIT) if each prime ideal over a nonzero principal ideal of R has height one. Let R be an integral domain with identity. A prime ideal P of R is called an associated prime ideal of a principal ideal in R if there exist some elements a, b ∈ R such that P is minimal over aR: bR = {x ∈ R|xb ∈ aR}. Consider an integral domain R with the following property: APIT: each associated prime ideal of a principal ideal in R has height one. One can easily show that R satisfies APIT if and only if R = ∩P∈X1(R)RP where X1(R) is the set of all height one prime ideals of R (cf. [6, Ex. 22, p.52]). The purpose of this paper is to show that R[X] satisfies APIT if and only if R satisfies APIT and R is an S-domain. (Recall that an integral domain R is an S-domain if for each height one prime ideal P of R, the expansion P [X] of P to R[X] has again height one.) All rings considered in this paper are commutative integral domains with identity.