I have this exercise that requires me to prove that two definitions are equal. The definitions regard principal nth root of unity and they read:
Let $\mathcal{R}$ be a commutative ring with $1$, let $n \in \mathbb{N}, n > 1$, be arbitrarily fixed, and let $ \omega \in \mathcal{R}$. Then $\omega$ is said to be a principal nth root of unity if:
(1) $\omega^n = 1$;
(2) $1 - \omega^k$ is not a zero divisor in $\mathcal{R}$ for all $1 \leq k < n$.
and
Let $\mathcal{R}$ be a commutative ring with $1$, let $n \in \mathbb{N}, n > 1$, be arbitrarily fixed, and let $ \omega \in \mathcal{R}$. Then $\omega$ is said to be a principal nth root of unity if:
(1) $\omega^n = 1$;
(2) $\sum_{j=o}^{n-1}{\omega^{jk} = 0}$ for all $1 \leq k < n$.
And although first assertion is same in the case of both definitions, I can't seem to be able to notice the relation between the second assertions. Can anyone please point me in the right direction on this matter?