What units are in the ring $\mathbb{Z}_2 \times \mathbb{Z}$?

Question

I'm stuck on this question.

I know how to find the units of rings for example, $\mathbb{Z}_7$ and $\mathbb{Z}_{15}$. I need to find elements that are relatively prime to $7$ and $15$. Having trouble with the product example.

Answer 1

Taking group of units preserves products, that's \begin{align} ((\Bbb Z/2\Bbb Z)\times\Bbb Z)^\times &=(\Bbb Z/2\Bbb Z)^\times\times\Bbb Z^\times\\ &=\{1+2\Bbb Z\}\times\{\pm 1\} \end{align} hence the units of $(\Bbb Z/2\Bbb Z)\times\Bbb Z$ are $(1+2\Bbb Z,\pm 1)$.