3928. MODULESthe ring elementsaiare nonzero and noninvertible, andaidividesajforij;Show that fork1,pk1M=pkMŠ.R=.p//mk.p/, wheremk.p/is the number ofaithat are divisible bypk. Concludethat the numersmk.p/depend only onMand not on the choiceof the direct sum decompositionMDA1˚A2˚˚As.(c)Show that the numbersmk.p/, aspandkvary, determinesandalso determine the ring elementsaiup to associates. Concludethat the invariant factor decomposition is unique.8.5.8.LetMbe a finitely generated torsion module over a PIDR. Letmbe a period ofMwith irreducible factorizationmDpm11pmss. Showthat for eachiand for allx2MOEpiŁ,pmiixD0.8.6. Rational canonical formIn this section we apply the theory of finitely generated modules of aprincipal ideal domain to study the structure of a linear transformation ofa finite dimensional vector space.
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