This preview shows page 1. Sign up to view the full content.
Unformatted text preview: 8.4. FINITELY GENERATED MODULES OVER A PID, PART I 375 s t = = D : It follows that A D U. s t = = I 1;i/A has .1;1/ entry equal to . The case that the nonzero entry is in the first row is handled similarly, with column operations rather than row opera tions. If divides all the entries in the first row and column, then row and column operations of type 1 can be used to replace the nonzero entries by zeros. n Remark 8.4.9. The proof of this lemma is nonconstructive, because in general there is no constructive way to find s and t satisfying s C t D . However, if R is a Euclidean domain, we have an alternative constructive proof. If divides , proceed as before. Otherwise, write D q C r where d.r/ < d./ . A row operation of the first type gives a matrix with r in the .i;1/ position. Then interchanging the first and i th rows yields a matrix with r in the .1;1/ position. Since d.r/ < d./ , we are done....
View
Full
Document
 Fall '08
 EVERAGE
 Algebra

Click to edit the document details