The first part of the paper will describe a recent result of K. Retert in (\cite{ Ret}) for $k[x_1,\ldots,x_n]$ and $k[[x_1,\ldots,x_n]]$. 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.
Categories: math.RA