Abstract <p>The article continues the study of multiplicatively idempotent semirings with the annihilator condition. It is proved that for multiplicatively idempotent semirings with zero the annihilator condition is equivalent to the equalizing property (Theorem 1). New conditions are obtained (the Rickart property, properties of a prime 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 are also proved, examples are given, and explanatory remarks are made.</p>