Torsion and divisibility in finitely generated commutative semirings

It is conjectured that (additive) divisibility is equivalent to (additive) idempotency in a finitely generated commutative semiring S. In this paper we extend this conjecture to weaker forms of these properties-torsion and almost-divisibility (an element a is an element of S is called almost-divisible in S if there is b is an element of Nsuch that b is divisible in S by infinitely many primes). We show that a one-generated semiring is almost-divisible if and only if it is torsion. In the case of a free commutative semiring F(X) we characterize those elements f is an element of F(X) such that for every epimorphism pi of F(X) torsion and almost-divisibility of pi(f) are equivalent in pi (F(X)).