Let $p$ be prime and let $\mathbb{Z}_{(p)} \subset \mathbb{Q}$ denote the set of all fractions $n/q$ for which $p \nmid q$. Is it true that $\mathbb{Z}_{(p)}$ is flat as a $\mathbb{Z}$-module?
Assume $M',M$ are $\mathbb{Z}$-modules and $f: M' \to M$ is injective, I want to prove that then $f \otimes 1: M' \otimes \mathbb{Z}_{(p)} \to M \otimes \mathbb{Z}_{(p)}$ is injective, i.e if $\sum a_i(m'_i \otimes n_i) \in M' \otimes \mathbb{Z}_{(p)}$ is an element such that $\sum a_i(f(m'_i) \otimes n_i)=0$, then the element is $0$, i.e the kernel is trivial.