Let be an integral domain. Prove that every maximal ideal in is irreducible.
We begin with a lemma. This lemma is actually true (with a minor modification) over a much larger class of rings. However, I am not entirely comfortable with the proof of the general version. I am, however, comfortable with this more specific version. I referred to a discussion at MathForum.org for a proof of the lemma.
Lemma: Let be an integral domain and let be an ideal which is finitely generated as a -module. If , then either or . Proof: Suppose . Let be a generating set for over ; in particular, some is nonzero. Since , there exist such that for each . In particular, for each , where is the Kronecker delta. That is, is a nontrivial solution to the matrix equation , where . Thinking of as embedded in its field of fractions, we have . On the other hand, by the Leibniz expansion of , we have , where (using the fact that is an ideal). In particular, , so that .
Now let be nonzero and maximal. If , then and , so that . If , then . By the lemma, either (a contradiction) or (also a contradiction). So either or is . Since is a unit in the semigroup of ideals of under ideal products, is irreducible.