Let $M$ be a flat $R$-module, where $R$ is a commutative ring and let $\phi:R \to S$ a ring homomorphism. Then, we have that $S$ is an $R$-module, where $rs=\phi(r)s$.
I want to prove that the canonical homomorphism $\phi:M \to M \otimes_R S$ is injective iff $\ker(\phi) \subseteq \mathrm{Ann}(M)= \lbrace r: rM=0 \rbrace$.
I can prove the assertion (=>), any idea about how I can prove the other hand?
Best Answer
Both directions can be proven if one observes that the map $m \mapsto m \otimes 1$ (which, by abuse of notation, was also denoted as "$\phi$") factors as $M \to M \otimes_R R/\mathrm{ker}(\phi) \to M \otimes_R S$ where the first map is injective (in fact, an isomorphism) if and only if $\mathrm{ker}(\phi) \subseteq \mathrm{Ann}(M)$, and the second map is injective because $R/\mathrm{ker}(\phi) \to S$ is injective and $M$ is flat.