Multiplicative inverses of $\Bbb Z \times \Bbb Z$

Question

This is more of a clarification.. I know the only multiplicative inverses of $\Bbb Z$ are $\{-1, 1\}$. I want to say that by the same principle, the only multiplicative inverses of $\Bbb Z \times \Bbb Z$ are $\{(-1, -1), (1, 1)\}$.

I can't seem to convince myself that $\{(-1, 1), (1, -1)\}$ should not be included here.. should they be?

TIA

Answer 1

Note that they are actually their own inverses: $(-1,1)\cdot (-1,1)=(1,1)$ which is the identity.

Answer 2

The unit group of a direct product is the direct product go its unit groups, see here:

Group of units of direct sum of rings is isomorphic to direct sum of the groups of units

So $$U(\Bbb{Z}\times \Bbb{Z})=U(\Bbb{Z})\times U(\Bbb{Z})=\{(\pm 1,\pm 1)\}.$$