which elements of $\mathbb{Z}_n$ give the entire ring?

Question

Which a ∈ $\mathbb{Z}_n$ satisfy 〈a〉$_\mathbb{Zn}$=$\mathbb{Z}_n$

From other theorems I'm thinking that $a_n$ must be a unit, but I don't know how to prove it... or if that is even correct.

Answer 1

Yes, there’s a very nice proof that says that the elements of $U(n)$ generate $\mathbb{Z}_n$

Answer 2

Assuming you mean $\langle a\rangle$ as an ideal in $\mathbb{Z}_{n}$.

Let's say $a \in \mathbb{Z}_{n}$ is a unit, so it has a multiplicative inverse $a^{-1}$. Then, $\langle a \rangle a^{-1} \subseteq \langle a \rangle$, so we have $1 \in \langle a \rangle$. But then $\langle a \rangle = \langle 1 \rangle = \mathbb{Z}_{n}$.