I noticed while studying for an introductory Algebra course, that the units $ x $ of $\Bbb {Z}_n, +, •$ had to be integers so that $\gcd(x, n) = 1$. But aren't those elements generators of $\Bbb {Z}_n, +$ too?
As far as I can tell, yes. But is there an underlying reason, theory or something to back this? Or is this just it?
The criterion is the same for each.
That $\operatorname {gcd}(a,n)=1$ is equivalent to $a$ being a unit, since by Bezout, this is when $\exists x,y$ such that $xa+yn=1$, or $x=a^{-1}$ exists.
The second fact is because $1$ generates $\Bbb Z_n,+$. And hence so does $a=a\cdot 1$ for any a such that $\operatorname{gcd}(a,n)=1$. That's because $\vert a\cdot 1\vert=\dfrac n{\operatorname {gcd}(a,n)}=n$.