Let D be a princple ideal domain and let . Prove that <p> is a maximal ideal in D iff p is irreducible.
Proof.
Assume p is irreducible, and let p = ab, then either
(Actually we should also specify that is not a unit also).
Here is half of the argument:
Say that is a maximal ideal and . Then . Thus either or . If the first case then and are associate elements so is a unit. If the second case then (because is maximal) and thus is a unit.