Can both additive and multiplicative operations in a semiring distribute over each other?

Question

In general, the multiplicative operation in a semiring distributes over the additive operation from both the left and the right. But i found some operations which satisfy all the conditions of a semiring structure in which the multiplicative operations distributes over the additive operation and vice-versa. Infact, monosemiring is the one where both the ring opera tions distribute over each other. Can i still call the set equipped with such operations other than monosemiring a semiring?

Answer 1

Such a semiring will be isomorphic to a boolean ring. It is easy to show that $a^2=a$ for all $a$ in such a semi ring. $a+0\times 0= (a+0)\times (a+0)$.