complete direct sum

The direct product of the rings R_γ, for γ some index set I, is the set product_(γ element I) R_γ = {f:I-> union _(γ element I) R_γ|f(γ) element R_γ all γ element I}. The ring direct product is confusingly also called the complete direct sum. The ring direct product, like the group direct product, has the universal property that if any ring X has a homomorphism to G and a homomorphism to H, then these homomorphisms factor through G×H in a unique way.