Prove that is an integral basis for . Find a basis (over ) for the ideal in such that , , and .
First we claim that and are linearly independent over . Indeed, if , then , and thus . So is a basis for (being a linearly independent set in the two dimensional -vector space ). We claim that every integer in is of the form where . Indeed, we have . So is an integral basis.
Next we claim that is a basis for over . Note that and . If , then evidently . Next, note that if , then , so that . That is, is a basis for over , and certainly is upper triangular.