{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

College Algebra Exam Review 365

# College Algebra Exam Review 365 - 8.4 FINITELY GENERATED...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 8.4. FINITELY GENERATED MODULES OVER A PID, PART I 375 s t ˇ=ı ˛=ı ˛ ˇ D ı : It follows that A D U. s t ˇ=ı ˛=ı I 1;i/A has .1;1/ entry equal to ı . The case that the nonzero entry ˇ 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 non-constructive, because in general there is no constructive way to find s and t satisfying s˛ C tˇ D ı . However, if R is a Euclidean domain, we have an alternative constructive proof. If ˛ divides ˇ , proceed as before. Otherwise, write ˇ 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

{[ snackBarMessage ]}

Ask a homework question - tutors are online