Prime elements in $\mathbb{Z}/n\mathbb{Z}$ I'm tring to determine the prime elements in the ring $\mathbb{Z}/n\mathbb{Z}$.   

Every ideal of $\mathbb{Z}/n\mathbb{Z}$ is of the form $m\mathbb{Z}/n\mathbb{Z}$, where $m$ is a divisor of $n$. And its residue ring is isomorphic to $\mathbb{Z}/m\mathbb{Z}$.
Hence a prime ideal of $\mathbb{Z}/n\mathbb{Z}$ is of the form $p\mathbb{Z}/n\mathbb{Z}$, where $p$ is a prime divisor of $n$.
Hence every prime element of $\mathbb{Z}/n\mathbb{Z}$ is of the form $p$ (mod $n$), where $p$ is a prime divisor of $n$.
