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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?

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    Oh, yes! It was obvio$u$s! How co$u$ld I be so silly!!!!! :) Sorry!2011-06-02

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