  The computational complexity of the circuit evaluation problem for finite
semirings is considered, where semirings are not assumed to have an additive or
multiplicative identity. The following dichotomy is shown: If a finite semiring
is such that (i) the multiplicative semigroup is solvable and (ii) it does not
contain a subsemiring with an additive identity $0$ and a multiplicative
identity $1 \neq 0$, then the circuit evaluation problem for the semiring is in
$\mathsf{DET} \subseteq \mathsf{NC}^2$. In all other cases, the circuit
evaluation problem is $\mathsf{P}$-complete.
