  Let $(R,\mathfrak{m}_R,k)$ be a one-dimensional complete local reduced
$k$-algebra over a field of characteristic zero. The ring $R$ is said to be
quasihomogeneous if there exists a surjection $\Omega_R\twoheadrightarrow
\mathfrak{m}$ where $\Omega_R$ denotes the module of differentials. We present
two characterizations of quasihomogeneity of $R$ in the situation when $R$ is a
domain: the first one on the valuation semigroup of $R$ and the other on the
trace ideal of the module $\Omega_R$.
