Let $\alpha_1, \dots, \alpha_N$ be the $2N$-th roots of $q$, i.e. $\alpha_i^{2N}\equiv 1 \bmod q$.
So, $\alpha_i$'s are roots of $x^N+1$.
Then, by Chinese Remainder Theorem (CRT),
$$\mathbb{Z}_q[x]/(x^N+1) \cong \mathbb{Z}_q[x]/(x-\alpha_1) \times \cdots \times \mathbb{Z}_q[x]/(x-\alpha_N)$$
where $q=1 \bmod 2N$.
Why $q=1 \bmod 2N$? If $q$ is not, is the above CRT not satisfied? Or do I know incorrect theorem? If I do, let me know the correct conditions and why the conditions are needed.