1.6.33 Fields, Groups, Rings

condition: After checking the axioms, determine whether the given algebra with two binary operations is a semiring or a ring. In this case: a) for a semiring (that is not a ring), check whether the semiring is commutative, idempotent, closed; b) for a ring, check whether it is Boolean, whether it has zero divisors, and whether the ring is a field. Given algebra: Set of matrices of the form \( \left(\begin{array}{ll}a & 0 \\ 0 & b\end{array}\right) \), where \( a, b \in\{0,1\} \), with addition and multiplication operations matrices, and the operations of addition and multiplication of elements are performed in the semiring \( \mathbb{B} \).