Multiplicatively Idempotent Semirings with Annihilator Condition

-The structural theory of semirings with additional conditions is considered. Multiplicatively idempotent semirings with the annihilator condition are studied. General properties of such semirings are considered. Examples are given. A criterion for the fulfillment of the annihilator condition in an arbitrary multiplicatively idempotent semiring with zero is proved (Proposition 6). In terms of annihilators, new abstract characterizations of semirings isomorphic to the direct product of a Boolean ring with unit and a Boolean lattice (Theorem 1) are obtained. The direct product of a Boolean ring and a distributive lattice with the annihilator condition is a multiplicatively idempotent semiring with the annihilator condition. Generally speaking, the opposite is not true (Theorem 2). An example of a general nature of a multiplicatively idempotent semiring with unit and with the annihilator condition, which is not isomorphic to the direct product of a Boolean ring and a distributive lattice, is constructed. In conclusion, a number of supplements are given.