Dedekind semidomains

Abstract: We define Dedekind semidomains as semirings in which each nonzero fractional ideal is invertible. Then we find some equivalent condition for semirings to being Dedekind. For example, we prove that a Noetherian semidomain is Dedekind if and only if it is multiplication. Then we show that a subtractive Noetherian semidomain is Dedekind if and only if it is a $\pi$-semiring and each of it nonzero prime ideal is invertible. We also show that the maximum number of the generators of each ideal of a subtractive Dedekind semidomain is 2.