  For several semirings S, two weighted finite automata with multiplicities in
S are equivalent if and only if they can be connected by a chain of
simulations. Such a semiring S is called "proper". It is known that the Boolean
semiring, the semiring of natural numbers, the ring of integers, all finite
commutative positively ordered semirings and all fields are proper. The
semiring S is Noetherian if every subsemimodule of a finitely generated
S-semimodule is finitely generated. First, it is shown that all Noetherian
semirings and thus all commutative rings and all finite semirings are proper.
Second, the tropical semiring is shown not to be proper. So far there has not
been any example of a semiring that is not proper.
