Ideals of proportionally modular numerical semigroups

A numerical semigroup S is an IPM-semigroup if there exists an ideal I of a proportionally modular numerical semigroup such that S=I∪{0}. Let S and S′ be numerical semigroups such that S⊆S′. We say that S′ is an ideal extension of S if S∖{0} is an ideal of S′. Clearly a numerical semigroup is an IPM-semigroup if and only if it admits an ideal extension that is a proportionally modular numerical semigroup. In this paper we characterize all the ideal extensions of an arbitrary numerical semigroup. We also give an algorithm to decide whether or not an arbitrary numerical semigroup is an IPM-semigroup.