2024/01/21 by Korbelář, Miroslav
#FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2401.11602
Let P be a finitely generated commutative semiring. It was shown recently that if P is a parasemifield (i.e. the multiplicative reduct of P is a group) then P cannot contain the positive rationals ℚ+ as its subsemiring. Equivalently, a commutative parasemifield P finitely generated as a semiring is additively divisible if and only if P is additively idempotent. We generalize this result using weaker forms of these additive properties to a broader class of commutative semirings in the following way. 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 cannot contain any non-finitely generated subring of ℚ.