Subalgebras of the polynomial algebra in positive characteristic and the   Jacobian
  Let $k$ be a field of characteristic $p>0$ and $R$ be a subalgebra of $k[X]=k[x_1,...,x_n]$. Let $J(R)$ be the ideal in $k[X]$ defined by $J(R)\Omega_{k[X]/k}^n=k[X]\Omega_{R/k}^n$. It is shown that if it is a principal ideal then $J(R)^q$ is a subalgebra of $R[x_1^p,...,x_n^p]$, where $q=p^n(p-1)/2$. 
A. V. Gavrilov
math.AC