In the proof of Theorem 4.3.6 the logic of the following red underlined sentence seems implausible.
I’m not sure whether it simply uses the following reasoning:
For a ring map $f: R\rightarrow S$, an $R$-module $M$, an $S$-module $N$ and an $R$-linear map $g: M\rightarrow N$, if the extension of scalars of $g$ $$S\otimes_R M\rightarrow N$$ is injective, then $g$ is injective.
This is obviously wrong since one can pick $f$ to be any non-injective ring map and $g: R\rightarrow S$ to be the homomorphism between regular module, then the extension of $g$ is the identity map, which is injective.
So my question is how to make sense of the underlined sentence in the picture. Does the condition suffice to deduce that the map is dominant?