  A Hausdorff topological semiring is called simple if every non-zero
continuous homomorphism into another Hausdorff topological semiring is
injective. Classical work by Anzai and Kaplansky implies that any simple
compact ring is finite. We generalize this result by proving that every simple
compact semiring is finite, i.e., every infinite compact semiring admits a
proper non-trivial quotient.
