This is a basic question on tensor products of linear maps. Let $R$ be a commutative ring and let $\varphi: M\to N$ be an injective linear map of finitely generated free $R$-modules.
Question: Are the $k$-th tensor power maps $\varphi^{\otimes k}: M^{\otimes k}\to N^{\otimes k}$ also injective for all $k\geq 0?$
To show this I've tried using the fact that $M$ and $N$ are flat $R$-modules, but I think I also need to assume that if $M$ and $N$ are flat then $M\otimes_R N$ is flat.
Is this the correct approach to take?
Many thanks in advance.
Hint: for a commutative ring $R$, the tensor product $M\otimes_RN$ of two free $R$-modules $M$ and $N$ is again a free module, hence flat.