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College Algebra Exam Review 382

College Algebra Exam Review 382 - 392(c 8 MODULES the ring...

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392 8. MODULES the ring elements a i are nonzero and noninvertible, and a i divides a j for i j ; Show that for k 1 , p k 1 M=p k M Š .R=.p// m k .p/ , where m k .p/ is the number of a i that are divisible by p k . Conclude that the numers m k .p/ depend only on M and not on the choice of the direct sum decomposition M D A 1 ˚ A 2 ˚ ˚ A s . (c) Show that the numbers m k .p/ , as p and k vary, determine s and also determine the ring elements a i up to associates. Conclude that the invariant factor decomposition is unique. 8.5.8. Let M be a finitely generated torsion module over a PID R . Let m be a period of M with irreducible factorization m D p m 1 1 p m s s . Show that for each i and for all x 2 MOEp i Ł , p m i i x D 0 . 8.6. Rational canonical form In this section we apply the theory of finitely generated modules of a principal ideal domain to study the structure of a linear transformation of a finite dimensional vector space.
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