Let be a Euclidean domain and let be a finitely generated (left, unital) -module. By this previous exercise, there is a surjective module homomorphism for some natural number . By this previous exercise, is finitely generated as an -module. If we let be an (ordered) basis for and let be an (ordered) generating set for , then for each , we have unique elements such that .
We now construct the matrix whose th row is simply the coefficients of . This is called a relations matrix with respect to the homomorphism , the basis , and the generating set .
- Show that interchanging and in the basis has the effect of interchanging the th and th columns of .
- Show that, for any , replacing the element in by () results in another basis. Show further that this has the effect of adding times the th column to the th column of .
It is clear that interchanging and in merely interchanges the th and th columns of , since the rows of are precisely the (ordered) expansions of the in terms of the .
Now we will show that replacing by results in a new basis. Recall that a basis is a linearly independent generating set. To see that is linearly independent, note that if , then for all since is linearly independent, and then . Now if , then since is a generating set, we have for some . Certainly then , so that is a generating set, and thus a basis, for .
Our argument that is a generating set now shows that the matrix for is obtained from that for by adding times column to column .