David wrote:

> Observe that $\mathbb{Z}_2$ is a proper one-dimensional subspace of the one-dimensional space $\mathbb{Z}$.

It's not a subspace, it's a quotient space. That is, there's no 1-1 homomorphism of modules $\mathbb{Z}_2 \to \mathbb{Z}$. Instead, there's an onto homomorhism of modules $\mathbb{Z} \to \mathbb{Z}_2$, sending each integer to that integer mod 2.

By the way, people call these things "modules", not "spaces", which connotes vector space. The word "dimension" is also not one we use for modules over a ring. So, nobody would say "the one-dimensional space $\mathbb{Z}$". They'd say "the free $\mathbb{Z}$-module of rank one, $\mathbb{Z}$".