# Wydawnictwa / Czasopisma IMPAN / Colloquium Mathematicum / Wszystkie zeszyty

## Some results on the kernels of higher derivations on $k[x, y]$ and $k(x, y)$

### Tom 122 / 2011

Colloquium Mathematicum 122 (2011), 185-189 MSC: Primary 13N15; Secondary 13A50. DOI: 10.4064/cm122-2-3

#### Streszczenie

Let $k$ be a field and $k[x, y]$ the polynomial ring in two variables over $k$. Let $D$ be a higher $k$-derivation on $k[x, y]$ and $\overline D$ the extension of $D$ on $k(x, y)$. We prove that if the kernel of $D$ is not equal to $k$, then the kernel of $\overline { D}$ is equal to the quotient field of the kernel of $D$.