Completely Prime Ideals in Multiplicatively Idempotent Semirings

Structure properties of multiplicatively idempotent semirings are considered. Basic results concerning completely prime ideals in multiplicatively idempotent semirings are obtained. The main theorem describes the commutative multiplicatively idempotent semirings with zero in which all completely prime ideals are maximal: up to isomorphism, such semirings are exhausted by direct products of a Boolean ring and a generalized Boolean lattice. Examples are given showing that the conditions of commutativity and the presence of zero are essential.