A characterization of general Z.P.I.-rings. II

Abstract

A commutative ring A is a general Z.P.I.-ring if each ideal of A can be represented as a finite product of prime ideals. We prove that a commutative ring A is a general Z.P.I.-ring if each finitely generated ideal of A can be represented as a finite product of prime ideals. We also give a characterization of Krull domains in terms of *-operations, as defined by Gilmer. Introduction. A commutative ring A is a iT-ring if each principal ideal of A can be represented as a product of prime ideals; A is a p-ring if each finitely generated ideal of A can be represented as a product of prime ideals. A commutative ring A is a general Z.P.I.-ring if each ideal of A can be represented as a product of prime ideals. Mori characterized nr-rings in [4], [5] and general Z.P.I.-rings in [6]. In [9] Wood extended Mori's results on nr-rings. Wood also gave a characterization of general Z.P.I.rings which is independent of Mori's work [10]. Let D be an integral domain with identity and with quotient field K. If F(D) is the set of nonzero fractional ideals of A, a mapping B--*B* of F(D) into F(D) is called a *-operation on D if the following three conditions hold for any a in K-{O}, and any B, C in F(D). (1) (a)*=(a), (aB)*=aB*. (2) BiB*, if Bc C, B*c C*. (3) (B*)*=B*. If (B)* (C)*, B and C are called *-equivalent, denoted B,,* C. If B=B*, B is called a *-ideal. An integral domain D is a Krull domain if D= n v, where { V} is a set of rank one discrete valuation rings with the property that for each nonzero x in D, xVa= Va for all but a finite number of the Va. In the first section of this paper, we give a new characterization of Krull domains. We prove that an integral domain A with identity is a Krull domain if and only if there is a *-operation on A such that each nonzero principal ideal of A is *-equivalent to a product of prime ideals. Then we consider nr-rings with no zero divisors, called ,r-domains. Theorem 1.2 states that A is a nr-domain with identity if and only if A is a Krull domain Received by the editors June 28, 1971. AMS 1970 subject class/ifcations. Primary 13F05, 13F10.