# Describe all prime and maximal ideals of $\mathbb{Z}_n$

I know that an ideal P in $\mathbb{Z}_n$ is prime if and only if $\mathbb{Z}_n/P$ is an integral domain and an ideal m in $\mathbb{Z}_n$ is maximal if and only if $\mathbb{Z}_n$/m is a field. I think I've figured out that $\mathbb{Z}_p$ where p is a prime that divides n make up the maximal ideals.

I have no idea how to figure out which are prime. Help?

Hint: Write $m=\Pi_{i=1}^{i=n}p_i^{n_i}$ where $p_i$ is a prime number, the prime and maximal ideals are generated by the class of $p_1^{n_1}..p_i^{n_i-1}..p_n^{i_n}$. The Chinese remainder theorem implies that $\mathbb{Z}_m\simeq \mathbb{Z}/p_1^{n_1}\times...\times \mathbb{Z}/p_1^{n_1}$ and $\mathbb{Z}_m/p_1^{n_1}..p_i^{n_i-1}..p_n^{i_n}\simeq\mathbb{Z}/p_i$.

• so what are the prime ideals? I don't really follow this – John Smith Dec 17 '17 at 2:31
• the prime and maximal are the same – Tsemo Aristide Dec 17 '17 at 2:33