$Im(f\otimes 1_M)=Im(f)\otimes M$ for flat module M

Question

Suppose $N,N'$ are both $A$ modules and $f:N\rightarrow N'$ is an $A$ module homomorphism and $M$ is a flat $A$ module. How does one show that $Im(f\otimes 1_{M})=Im(f)\otimes M$, where $f\otimes 1_{M}:N\otimes M\rightarrow N'\otimes M$?

Answer 1

Break up $f$ as $f_1\circ f_2$ where $f_2:N\to \text{Im}\, f$ is the restriction of $f$ and $f_2:\text{Im}\, f\to N'$ is the inclusion. Then $f_1$ and $f_1\otimes 1_M$ are surjective, so that the image of $f\otimes 1_M$ is the same as that of $f_2\otimes 1_M$. As $M$ is flat, tensoring with $M$ preserves injections: so that $f_2\otimes 1_M$ is an injective map from $(\text{Im}\,f)\otimes 1_M$ to $N'\otimes M$ with image $\text{Im}(f\otimes 1_M)$. Therefore $(\text{Im}\,f)\otimes 1_M$ and $\text{Im}(f\otimes 1_M)$ are isomorphic