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 described 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 ³ 2/ 4 .x ³ 1/ and g.x/ D .x ³ 2/ 2 .x ³ 1/ 2 ± x 2 C 1 ² 3 : The factorizations displayed for f.x/ and g.x/ are the irreducible factorizations in Q ŒxŁ . Let M denote the QŒxŁ –module M D Q ŒxŁ=.f / ˚ Q ŒxŁ=.g/ . Then M Š Q ŒxŁ=. .x ³ 2/ 4 / ˚ Q ŒxŁ=. .x ³ 1// ˚ Q ŒxŁ=. .x ³ 2/ 2 / ˚ Q ŒxŁ=. .x ³ 1/ 2 / ˚ Q ŒxŁ=. .x 2 C 1/ 3 /...
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 Fall '08
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 Algebra, Mathematical terminology, Module theory

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