The Coproduct of Ideals and the Coprime Spectrum

We introduce the coproduct of ideals and the notion of coprime ideal, extending the Heyting-algebra perspective on ideal lattices to arbitrary commutative rings. The resulting coprime spectrum is a topological space, defines a covariant functor on epimorphisms of commutative rings that do not annihilate coprime ideals, classifies fields among integral domains, and in dual rings is homeomorphic to the prime spectrum. This offers an elementary, lattice-theoretic counterpart to the geometric study of nilpotents.