| Let R and S be commutative rings, not necessarily with identity. We investigate the ideals, prime ideals, radical ideals, primary ideals, and maximal ideals of R x S. Unlike the case where R and S have an identity, an ideal (resp., primary ideal, maximal ideal) of R x S need not be a "subproduct" I x J of ideals. We show that for a ring R, for each commutative ring S every ideal (resp., primary ideal, maximal ideal) is a subproduct if and only if R is an e-ring (i.e., for r ∈ R, there exists er ∈ R with err = r) (resp., u-ring (i.e., for each proper ideal A of R, A ≠ R), the abelian group (R/R2, +) has no maximal subgroups).;Let R be a commutative ring not necessarily having an identity. Then R is a general ZPI-ring if every ideal of R is a product of prime ideals. S. Mori showed that a general ZPI-ring without identity is either (1) an integral domain, (2) a ring R where every ideal of R including 0 is a power of R, (3) K x R where K is a field and R is a ring as in (2), or (4) K x D where K is a field and D is a domain with every nonzero ideal of D a power of D. We prove that if R is a ring as in (2), then there is an SPIR S with S = R[1] having R as its maximal ideal. Moreover, there is a complete DVR (D, (pi)) with D = (pi)[1] so that S and R are homomorphic images of D and (pi), respectively. |
