Simplicity and Commutative Bases of Derivations in Polynomial and Power Series Rings

Abstract

The first part of the paper will describe a recent result of K. Retert in (\cite{Ret}) for $k[x_1,\dots ,x_n]$ and $k[[x_1,\dots ,x_n]]$. This result states that if $\mathfrak{D}$ is a set of commute $k$-derivations of $k[x,y]$ such that both ${\partial }_x\in \mathfrak{D}$ and the ring is $\mathfrak{D}$-simple, then there is $d\in \mathfrak{D}$ such that $k[x,y]$ is $\{{\partial }_x,d\}$-simple. As applications, we obtain relationships with known results of A. Nowicki on commutative bases of derivations.