Primary ideals and valuation ideals

Abstract

Introduction. Let D be an integral domain with identity. In [2], Gilmer and Ohm considered the problem of characterizing domains D such that the set &2(D) of primary ideals of D is a subset of the set Y(D) of valuation ideals of D. If the ascending chain condition (a.c.c.) for prime ideals holds in D, then &2(D) ' Y(D) if and only if D is a Prilfer domain; a domain in which primary ideals are valuation ideals need not be Priufer if the assumption concerning the a.c.c. for prime ideals is dropped (see Theorem 3.8 and ?5 of [2]). In case a.c.c. for prime ideals does not hold in D, Gilmer and Ohm left open the question as to when primary ideals of D are valuation ideals. In [1], Gilmer showed that the question as to whether &2(D)cY'(D) or not is closely related to the structure of the set of prime ideals of D. Before mentioning these results and their relation to this paper, we introduce some terminology. Let R be a commutative ring with identity, let P be a prime ideal of R, and let {Q} be the set of P-primary ideals of R. We consider the following conditions: I. {Q,} is linearly ordered under c II. M= n. Qc, is a prime ideal. III. There are no prime ideals of R properly between M and P. IV. If P1 is any prime ideal of R properly contained in P, then P1 c M. Following [1], we say P is an S-ideal if I, II, and IV hold. It is clear that IV implies III. If I-III hold, we say P is a weak S-ideal. R is an S-ring if each prime ideal of R is an S-ideal; weak S-ring is defined analogously. Corollary 2.4 of [1] shows that if D is an S-domain, then &2(D) Y'(D). The proof of Corollary 2.4 does, in fact, show that in a weak S-domain primary ideals are valuation ideals. The status of the converse of Corollary 2.4 was considered in [1], but was not determined. In ?3 we prove that for P prime in D, each P-primary ideal is a valuation ideal if and only if conditions I and II hold. We thereby obtain what we feel is a satisfactory characterization of domains in which primary ideals are valuation ideals. To resolve the questions of whether the condition &(D)cY'(D) implies D is an Sdomain or a weak S-domain we need only determine in the global case whether condition IV or condition III depends upon I and II. These questions are answered in ?5. Example 5.8 shows III does not depend on I and II and hence IV does not