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.
