Canonical map is injective

Question

Let $A$ and $B$ be commutative rings, and let $f:A\to B$ be a faithfully flat ring homomorphism. How can I show that for any $A$-module $M$, the canonical map $M\to M\otimes_AB$ is injective?

I was wondering that do we need the assumption that $f$ is faithfully flat as the part (c) of the problem does not use the mapping $f$ so is the assumption only for other parts of the problem?

(Liu, Algebraic Geometry and Arithmetic Curves, Exercise 1.2.19(c).)

Answer 1

Let $K=\ker(M\to M\otimes_AB)$. Then $K=\{x\in M:x\otimes 1=0\}$. We show that $K\otimes_AB=0$: $x\otimes b=(x\otimes 1)b=0$, so $K=0$.