Root of unity is a unit in a ring

Question

Wikipedia states that any root of unity in a ring is a unit(meaning it has a multiplicative inverse). Specifically if $r^n=1$, then $r^{n-1}$ is a multiplicative inverse of $r$.

But this is not true since in the ring of integers, except for $1$ and $-1$, any number is a root of unity, e.g. $5^0=1$, but there is no multiplicative inverse for the number $5$.

Am I understanding this definition correctly?

Answer 1

No, you are not. An element $r$ of a ring $R$ is a root of unity if $r^n=1$ for some $n\in\mathbb N=\{1,2,3,\ldots\}$. $\ \ \ \$

Answer 2

Well, for each element $r$ in a ring $R$, $r^0=1$.

However, you refer to the case that $r^n=1$ for some $n>0$. Then, indeed, $$r\cdot r^{n-1} = 1 = r^{n-1}\cdot r$$ and so $r^{n-1}$ is the inverse of $r\ne 0$.