Suppose $R$ is a commutative Artinian local ring over an algebraically closed characteristic 0 field $k$. Suppose $f\in R$ is such that $df=0$ (in the sense that the element $df$ vanishes in the module of Kahler differentials). Is $f$ necessarily in $k$?

EDIT: Forgot to include $k$ is algebraically closed.

Suppose $R$ is a commutative Artinian local ring over an algebraically closed characteristic 0 field $k$. Suppose $f\in R$ is such that $df=0$ (in the sense that the element $df$ vanishes in the module of Kähler differentials). Is $f$ necessarily in $k$?

Suppose $R$ is a commutative Artinian local ring over a characteristic 0 field $k$. Suppose $f\in R$ is such that $df=0$ (in the sense that the element $df$ vanishes in the module of Kahler differentials). Is $f$ neccessarily 0?

Suppose $R$ is a commutative Artinian local ring over an algebraically closed characteristic 0 field $k$. Suppose $f\in R$ is such that $df=0$ (in the sense that the element $df$ vanishes in the module of Kahler differentials). Is $f$ necessarily in $k$?

EDIT: Forgot to include $k$ is algebraically closed.

Suppose $R$ is a commutative Artinian local ring over a characteristic 0 field $k$. Suppose $f\in R$ is such that $df=0$ (in the sense that the element $df$ vanishes in the module of Khaler differentials). Is $f$ neccessarily 0?

Suppose $R$ is a commutative Artinian local ring over a characteristic 0 field $k$. Suppose $f\in R$ is such that $df=0$ (in the sense that the element $df$ vanishes in the module of Kahler differentials). Is $f$ neccessarily 0?