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College Algebra Exam Review 380

College Algebra Exam Review 380 - submodules C i;j agree...

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390 8. MODULES using Theorem 8.5.9 , and then apply Corollary 8.5.8 to each of the primary components. Alternatively, first apply the invariant factor decomposition to M , exhibiting M as a direct sum of cyclic modules. Then apply the pri- mary decomposition to each cyclic module; by Lemma 8.5.11 , one obtains a direct sum of cyclic modules with period a power of an irreducible. For uniqueness, suppose that f C i W 1 ± i ± K g and f D i W 1 ± i ± L g are two families of cyclic submodules of M , each with period a power of an irreducible, such that M D C 1 ˚ ²²² ˚ C K and M D D 1 ˚ ²²² ˚ D L . Let m be a period of M with irreducible factorization m D p m 1 1 ²²² p m s s . Then for each of the cyclic submodules in the two families has period a power of one of the irreducibles p 1 ;:::;p s . Relabel and group the two families accordingly: f C i g D [ p j f C i;j W 1 ± i ± K.j/ g ; and f D i g D [ p j f D i;j W 1 ± i ± L.j/ g ; where the periods of C i;j and D i;j are powers of p j . It follows from the previous lemma that for each j , K.j/ M i D 1 C i;j D L.j/ M i D 1 D i;j D MŒp j Ł: Corollary 8.5.8 implies that K.j/ D L.j/ and the annihilator ideals of the
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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 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 factor-izations 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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