Torsion factors of commutative monoid semirings

Let P be a finitely generated commutative semiring with a unity. It was shown recently that if the multiplicative reduct of P is a group then P is additively idempotent. We extend this result by showing that P is additively idempotent, provided that P is additively divisible. We further generalize this result using a weaker form of divisibility (almost-divisibility) as follows. Let S be a semiring that is a factor of a monoid semiring N[C] where C is a submonoid of a free commutative monoid of finite rank. Then the semiring S is additively almost-divisible if and only if S is torsion. In particular, we show that if S is a ring then S contains no non-finitely generated subring of Q.