Show every prime ideal in $\mathbb{Z}$ is of form $\langle p\rangle$ where $p$ is prime.

Question

An ideal $P$ is called a prime ideal if whenever $ab\in P$ we have either $a \in P$ or $b \in P$. I am confused about how to connect the definition with with $\langle p\rangle$.

Answer 1

Hint: use the well-ordering principle to find a candidate $p$ to serve as the generator for your ideal. Then use the euclidean algorithm to show every element in the ideal must be a multiple of $p$.

Answer 2

$\mathbb{Z}$ is a principal ideal domain.

This follows since it's a Euclidean domain; it has about a million answers here.

Therefore the ideals are precisely $\langle n \rangle$ for $n \in \mathbb{N}^{\geq 0}$.

The maximal ideals are precisely $\langle p \rangle$ for $p$ prime.

Proof: Since if $\langle pn \rangle$ is an ideal with $p$ prime and $n$ a non-zero nonunit integer, then $\langle p \rangle \supset \langle pn \rangle$; conversely, if $\langle p \rangle \subseteq \langle m \rangle$ then $m \mid p$ so $m=p$ or $m=1$.