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Unformatted text preview: submodules C i;j agree with those of the submodules D i;j up to order. It follows that K D L and that the list of annihilator ideals of the submodules C i agree with the list of annihilator ideals of the submodules D i , up to order. n The periods p n i;j j of the direct summands in the decomposition de-scribed in Theorem 8.5.16 are called the elementary divisors of M . They are determined up to multiplication by units. Example 8.5.17. Let f.x/ D .x 2/ 4 .x 1/ and g.x/ D .x 2/ 2 .x 1/ 2 x 2 C 1 3 : The factorizations displayed for f.x/ and g.x/ are the irreducible factor-izations in Q x . Let M denote the Qx module M D Q x=.f / Q x=.g/ . Then M Q x=. .x 2/ 4 / Q x=. .x 1// Q x=. .x 2/ 2 / Q x=. .x 1/ 2 / Q x=. .x 2 C 1/ 3 /...
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- Fall '08