A note on $p$-bases of rings

Let $R\supseteq R'\supseteq R^p$ be a tower of rings of characteristic $p>0$. Suppose that $R$ is a finitely presented $R'$-module. We give necessary and sufficient conditions for the existence of $p$-bases of $R$ over $R'$. Next, let $A$ be a polynomial ring $k[X_1,\dots ,X_n]$ where $k$ is a perfect field of characteristic $p>0$, and let $B$ be a regular noetherian subring of $A$ containing $A^p$ such that $[Q(B)\, :\, Q(A^p)]=p$. Suppose that $Der_{A^p}(B)$ is a free $B$-module. Then, applying the above result to a tower $B\supseteq A^p\supseteq B^p$ of rings, we shall show that a polynomial of minimal degree in $B-A^p$ is a $p$-basis of $B$ over $A^p$.