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 $k[X]=k[x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n]$. Let $J(R)$ be the ideal in $k[X]$ defined by $J(R){\mathrm{\Omega }}_{k[X]\mathrm{/}k}^n=k[X]{\mathrm{\Omega }}_{R\mathrm{/}k}^n$. It is shown that if it is a principal ideal then $J(R{)}^q$ is a subalgebra of $R[x_1^p,\mathrm{.}\mathrm{.}\mathrm{.},x_n^p]$, where $q=p^n(p-1)\mathrm{/}2$.