Does flat ring homomorphism send non-zero divisors to non-zero divisors?

Question

Suppose $f:A\rightarrow B$ is a flat ring homomorphism, and suppose $M$ is an $A$-module, with $a\in A$ a non-zero divisor of $M$. Further suppose $M/aM \otimes B \neq 0$. Is it true that $f(a)$ is a non-zero divisor of $M\otimes B$?

Answer 1

We know $m \in M \longmapsto am \in M$ is injective, so since $B$ flat over $a$, the $B$-endomorphism $aId \otimes Id$ of $M \otimes B$ is injective. Now, this endomorphism is exactly the multiplication by $f(a)$.