Ideals in direct products of commutative rings

Abstract

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 × S. Unlike the case where R and S have an identity, an ideal (or primary ideal, or maximal ideal) of R × S need not be a 'subproduct' I × J of ideals. We show that for a ring R, for each commutative ring S every ideal (or primary ideal, or maximal ideal) is a subproduct if and only if R is an e-ring (that is, for r ∈ R, there exists er ∈ R with err = r) (or u-ring (that is, for each proper ideal A of R, √A ≠ R)), the Abelian group (R/R2 ,+) has no maximal subgroups). © 2008 Australian Mathematical Society.