Families of k-derivations on k-algebras

Let $A$ be an integral $k$-algebra of finite type over a field $k$ of characteristic zero. Let ${\cal{F}}$ be a family of $k$-derivations on $A$ and $M_{\cal{F}}$ the $A$-module spanned by ${\cal{F}}$. In this paper, we generalize a result due to A. Nowicki and construct an element $\partial$ of $M_{\cal{F}}$ such that $\ker \partial=\cap_{d\in {\cal{F}}} \ker d$. Such a derivation is called ${\cal{F}}$-minimal. Then we establish a density theorem for ${\cal{F}}$-minimal derivations in $M_{\cal{F}}$.