Let be an algebraic number field with ring of integers . Fix an ideal . Recall that, for an ideal , . Prove the following: (1) is an -submodule of , (2) If is an ideal, then is an ideal, and (3) If are ideals, then .
Let and let . Now for all , we have , since and (because is an ideal). Since , by the submodule criterion, is an -submodule of .
Suppose . Note that if , then where and . Certainly then , as desired in (3).
Suppose ; by the previous argument, we have . Suppose and , with and . Certainly then since is an ideal we have . Since , by the submodule criterion, is an ideal in .