Multiplicatively idempotent semirings with annihilator condition

The article is devoted to the structural theory of semirings with additional conditions. It studies multiplicatively idempotent semirings with the annihilator condition. General properties of such semirings are considered. The work is accompanied by examples. 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, the authors give new abstract characterizations of semirings isomorphic to the direct product of a Boolean ring with identity and a Boolean lattice (Theorem 1). The direct product of a Boolean ring and a distributive lattice with the annihilator condition is a multiplicatively idempotent semiring with the annihilator condition. The converse assertion is not true in general (Theorem 2). The article presents an example of the general nature of a multiplicatively idempotent semiring with identity and with the annihilator condition, which is not isomorphic to a direct product of a Boolean ring and a distributive lattice. A number of additions complete the research.