What is $\mathbb{Z}_1^\times$?

Question

In general $\mathbb{Z}_n^\times$ is the group of units of $\mathbb{Z}_n$. So I see that $\mathbb{Z}_8^\times = \{1, 3, 5, 7\}$. In general I see that one just finds the elements that are relatively prime with $n$.

My question is: What is $\mathbb{Z}_1^\times$?

I see that $\mathbb{Z}_1 = \{0\}$ and that in this group $0=1=2=3=4=\dots$. And since this is a group under addition, I would guess that $\mathbb{Z}_1^\times = \mathbb{Z}_1 = \{0\}$.

Answer 1

Your guess is right: in the trivial ring, every element is a unit, so $\mathbb{Z}_1^\times=\mathbb{Z}_1$.