  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 K\"ahler 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 geometric algorithm to construct all possible semimodules in this
subset for a given value semigroup.
