Multiplicatively Idempotent Semirings in which All Congruences Are Ideal

The study of multiplicatively idempotent semirings with additional conditions is continued. It is proved that every multiplicatively idempotent semiring with ideal congruences is isomorphic to the direct product of a Boolean ring and a generalized Boolean lattice. Thus, a new abstract characterization is obtained for the direct products of Boolean rings and generalized Boolean lattices. Examples are given.