There is given the ring $R := k[T_1, T_2, ..., T_n]/(f_1, ..., f_r)$ of finite type, a field extension $k \subset k'$ and $R' := R \otimes _k k' \cong k'[T_1, T_2, ..., T_n]/(f_1, ..., f_r)$. How can I prove that $R'$ is a faithfully flat $R$-module?
My ideas (flatness):
If $M, N$ are $R$-modules with an injection $\phi: M \to N$ I have to show that the induced module morphism $ \bar{\phi}: M \otimes_R R' \to N \otimes_R R'$ is injective, too. According to the definition of $R'$ we have $M \otimes_R R' \cong M \otimes_R k'$ and for N vice versa. So have to show that that $ M \otimes_R k' \to N \otimes_R k'$ is injective or respectively that the functor $ - \otimes_R k'$ between $R$-modules is exact but I unsure that if it's true ...
Your mistake is that you say $- \otimes_R R' = - \otimes_R k'$, but actually we have $$- \otimes_R R' = - \otimes_k k'$$ and the latter is clearly a faithful exact functor since field extension are always faithfully flat.