In this paper we characterize the value set \(\Delta \) of the \(R\) -modules of the form \(R+zR\) for the local ring \(R\) associated to a germ \(\xi \) of an irreducible plane curve singularity with one Puiseux pair. In the particular case of the module of Kähler differentials attached to \(\xi \) , we recover some results of Delorme. From our characterization of \(\Delta \) we introduce a proper subset of semimodules over the value semigroup of the ring \(R\) . Moreover, we provide a combinatorial algorithm to construct all possible semimodules in this subset for a given value semigroup.
