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 $\langle p\rangle$.
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This fact is very useful: it is basically that "the maximal ideals are precisely the prime ideals in a principal ideal domain". – Patrick Stevens Apr 05 '17 at 07:32
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1@user26857 Very true. I can't edit my comment now, but I should say "the nonzero prime ideals". – Patrick Stevens Apr 05 '17 at 07:46
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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 $p$.
Alain
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$\mathbb{Z}$ 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 $m=1$.
Patrick Stevens
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