ON GENERAL ZPI-RINGS

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 with identity, it is clear that R is a Dedekind domain1 M. Sono [14] and E. Noether [13] were among the first to consider Dedekind domains. For a historical development of the theory of Dedekind domains see [4; pp. 31-32]. 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. S. Mori in [11] considered both general Z.P.I.-rings with identity and Z.P.I.-rings without identity which contain no proper zero divisors, but Mori's results in these cases are not as sharp as those of Asano and Gilmer. 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. © 1969 by Pacific Journal of Mathematics.