How can I prove that $2\mathbb{Z}$ is NOT a direct summand of $\mathbb{Z}$, as $\mathbb{Z}$-modules?

Question

I know that $\mathbb{Z}=2\mathbb{Z}+\{0\}$ (with "+" I mean "direct sum"). Is this enough to prove that $2\mathbb{Z}$ is NOT a direct summand of $\mathbb{Z}$, as $\mathbb{Z}$-modules?

Answer 1

Check that for any

$$\;m\Bbb Z\le\Bbb Z\;\;,\;\;2\Bbb Z\cap m\Bbb Z\neq0$$

and thus $\;2\Bbb Z\;$ , or for that matter any non-trivial subgroup of the integers, cannot be a non-trivial direct summand.