Multiplicatively idempotent semirings with annihilator condition. II

The article continues the study of multiplicatively idempotent semirings with the annihilation condition. It is proven that for multiplicatively idempotent semirings with zero the annihilation condition is equivalent to the equalizing property (Theorem 1). New conditions are obtained (Rickart property, properties of a simple spectrum, and others) under which a multiplicatively idempotent semiring is isomorphic to the direct product of a Boolean ring and a generalized Boolean lattice (Theorems 2 and 3). Some other statements have also been proved, examples have been given, and explanatory remarks have been made.