  Let $k$ be a field of characteristic zero. This paper studies a problem
proposed by Joseph F. Ritt in 1950. Precisely, we prove that (1) If $p\geq 2$
is an integer, for every integer $i\in\mathbb{N}$, the nilpotency index of the
image of $T_i$ in the ring $k\{T\}/[T^p]$ equals $(i+1)p-i$. (2) For every pair
of integers $(i,j)$, the nilpotency index of the image of $T_iU_j$ in the ring
$k\{T\}/[TU]$ equals $i+j+1$.
