Subring of $\mathbb{Z} \times \mathbb{Z}$

Question

Let $R = \mathbb{Z} \times \mathbb{Z}$ and let $S$ be a subring of $R$ isomorphic to $R$. Show that $S = R$.

I know that as $S$ is subring, $S \subseteq R$. But after this I only get that $A$ is isomorphic to a subset of $B$, but I think this is not enough.

Could you give me any idea to finish this? Thanks in advance!

Answer 1

Since $S \simeq R$, they have the same sets of idempotents: $(0,1)$, $(1,0)$ and $(1,1)$ (and zero). Hence $S$, as a subring, has to contains those elements — they are the only nonzero idempotents in $R$. But $R$ is generated by those three elements, so $S = R$.