IDEALS IN DIRECT PRODUCTS OF COMMUTATIVE RINGS

Source abstract

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 e r ∈ R with e r r = r ) (or u -ring (that is, for each proper ideal A of R , $\sqrt {A}\not =R$ )), the Abelian group ( R / R 2 ,+) has no maximal subgroups).