Two Criteria For Quasihomogeneity

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 ${\mathrm{\Omega }}_R\twoheadrightarrow \mathfrak{m}$ where ${\mathrm{\Omega }}_R$ 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 ${\mathrm{\Omega }}_R$.