How to show fraction field is flat (without localization)

To prove that an $R$-module $M$ is flat, it suffices to show that for every ideal $I \subset R$, the canonical map $I \otimes_R M \rightarrow M$ is injective.

When $R$ is a domain and $M$ is the field of fractions of $R$, we have that every element of $I \otimes_R M$ is expressible as a simple tensor, that is, $i \otimes m$ for some $i \in I$ and $m \in M$. This is a pleasant exercise in finding a common denominator.

With this, we must show that $i \otimes m \rightarrow im$ is an injective map. But $im = 0$ if and only if $i = 0$ or $m = 0$ in the field of fractions of $R$, and this is if and only if $i \otimes m = 0$.