Let be the ring of integers in an algebraic number field . Suppose is an ideal and an element. Show that there exists an ideal such that .
Write as a product of maximal ideals, and let . Note that as subsets of , we have . (Using Lemma 8.23 in TAN.) On the other hand, since , is an ideal of , as we argue. By definition, . If and , where and , then . Since , by the submodule criterion is an ideal of .