Let $R$ be a ring an let $A$ and $B$ be two $R$-algebras. If $S$ is faithfully flat over $R$ then we say that $A$ is a $S$-twisted form of $B$ if $A\otimes_R S$ and $B\otimes_R S$ are isomorphic as $S$-algebras.
Now, if $F$ is a free $R$-module, $ rank R=n$, and $S$ is a commutative algebra which is faithfully flat over the commutative and unitary ring $R$, could you prove that a $R$-module $P$ is a $S$-twisted form of $F$ if and only if $P\otimes_R S$ is a free $S$-module of rank n?