Two criteria for quasihomogeneity

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