Existence of elementary embedding of the integers into $\bigoplus_p {\bf Z}_{(p)}$

Question

For every prime $p$ let ${\bf Z}_{(p)}$ be the subgroup of $({\bf Q}, 0, +, -)$ consisting of fractions $a/b$ whose denominator is not divisible by $p$.

Let $Z = \prod_p {\bf Z}_{(p)}$ where the product is finite support.

Is there an elementary embedding ${\bf Z} \to Z$?

I think the answer is no.

Answer 1

Hint: Let $f:\mathbb{Z}\to Z$ be any homomorphism. Show that there must be some $n>1$ such that $f(1)$ is divisible by $n$ in $Z$, and use this to show $f$ is not an elementary embedding.