Vol. 30, No. 3, 1969

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On general Z.P.I.-rings

Craig A. Wood

Vol. 30 (1969), No. 3, 837–846

A commutative ring in which each ideal can be expressed as a finite product of prime ideals is called a general Z.P.I.-ring (for Zerlegungsatz in Primideale). A general Z.P.I.-ring in which each proper ideal can be uniquely expressed as a finite product of prime ideals is called a Z.P.I.-ring. Such rings occupy a central position in multiplicative ideal theory. In case R is a domain wilh identity, it is clear that R is a Dedekind domainl and the ideal theory of R is well known. If R is a domain without identity, the following result of Gilmer gives a somewhat less known characterization of R: If D is an integral domain without identity in which each ideal is a finite product of prime ideals, then each nonzero ideal of D is principal and is a power of D; the converse also holds. Also somewhat less known is the characterization of a general Z.P.I.-ring with identity as a finite direct sum of Dedekind domains and special primary rings.2

This paper considers the one remaining case: R is a general Z.P.I.-ring with zero divisors and without identity. A characterization of such rings is given in Theorem 2. This result is already contained ih a more obscure form in a paper by S. Mori. The main contribution here is in the directness of the approach as contrasted to that of Mori.

Mathematical Subject Classification
Primary: 13.50
Received: 5 November 1968
Published: 1 September 1969
Craig A. Wood