depend on I and II. Example 5.9 shows IV is independent of I-III-that is, a weak ■S-domain need not be an S-domain. We begin in §2 by considering a prime ideal 7 of a commutative ring with identity such that the set of 7-primary ideals is chained under £. All rings considered in this paper are assumed to be commutative and we assume that each integral domain considered has an identity. The terminology used is that of Zariski-Samuel [6], [7]. We shall make frequent use of the material from [6]

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... quotient rings of, in our case, an integral domain. We shall also use the fact that in considering i;-ideals of a domain D, where v is a valuation nonnegative on D, there is no loss of generality in assuming that D and the valuation ring of v have the same quotient field [7, p. 340]. While we feel that a reader can understand most of the results of this paper without having read [2], it is unlikely that he can appreciate its results independent of [1] and [2]. 2. Linearly ordered systems of primary ideals. Throughout this section P denotes a prime ideal of R, a commutative ring with identity, and {Qa} denotes the set of 7-primary ideals. We shall assume that the set {Qa} is linearly ordered under £ ; we denote by M the intersection of the ga's. 2.1. Lemma. If(Qe} is a subset of{Qa} such that B=(~\B QB^M, then y/B=P. Equivalently, if\/B<=P, then B=M. Proof. Since B=>M, there exists a 7-primary ideal Q such that B^Q. Thus QB$Q for any ß so that ô=ô/s for each ß, and consequently, Q^(~)B Qe = B. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use