Flatness of finitely generated (/finitely presented) module carries to power series module?

Question

Let $M$ be a finitely generated flat module over a commutative ring with unity $R$ . Then is $M[[x]]$ also flat over $R[[x]]$ ? If this is not always true , then what if we also assume $M$ is finitely presented and flat over $R$ ; is $M[[x]]$ flat over $R[[x]]$ then ?

Answer 1

I think this is true. Let $0\to K\to N$ be an inclusion of $R[[x]]$ modules. Then, $0\to K\otimes_R M\to N\otimes_R M$ is exact, since $M$ is flat over $R$. Now, $K\otimes_R M=K\otimes_{R[[x]]} R[[x]]\otimes_R M=K\otimes_{R[[x]]} M[[x]]$ and similarly for $N$, proving flatness of $M[[x]]$ over $R[[x]]$.