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In this paper we characterize the value set $\Delta$ of the $R$-modules ofthe form $R+zR$ for the local ring $R$ associated to a germ $\xi$ of anirreducible plane curve singularity with one Puiseux pair. In the particularcase of the module of K\"ahler differentials attached to $\xi$, we recover someresults of Delorme. From our characterization of $\Delta$ we introduce a propersubset of semimodules over the value semigroup of the ring $R$. Moreover, weprovide a geometric algorithm to construct all possible semimodules in thissubset for a given value semigroup.

Curve singularities with one Puiseux pair and value sets of modules over their local rings