For which additive submonoids M of nonnegative rationals is F[X;M] AP?

We characterize the submonoids $M$ of the additive monoid $\Q_+$ of nonnegative rational numbers for which the irreducible and the prime elements in the monoid domain $F[X;M]$ coincide. We present a diagram of implications between some types of submonoids of $\Q_+$, with a precise position of the monoids $M$ with this property.