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Unformatted text preview: R as R modules. In the remaining exercises, R denotes a principal ideal domain. 8.4.5. Let M be a a free module of rank n over R . Let N be a submodule of M . Suppose we know that N is nitely generated (but not that N is free). Adapt the proof of Theorem 8.4.12 to show that N is free. 8.4.6. Let V and W be free modules over R with ordered bases .v 1 ;v 2 ;:::;v n / and .w 1 ;w 2 ;:::;w m / . Let ' W V ! W be a module homomorphism. Let A D .a i;j / be the m by n matrix whose j th column is the coordinate vector of '.v j / with respect to the ordered basis .w 1 ;w 2 ;:::;w m / , '.v j / D X i a i;j w j : Show that for any element P j x j v j of M , '. X j x j v j / D w 1 ;:::;w m A 2 6 4 x 1 : : : x n 3 7 5 :...
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 Fall '08
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 Algebra, Matrices

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