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College Algebra Exam Review 379

# College Algebra Exam Review 379 - submodules AŒp j Ł...

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8.5. FINITELY GENERATED MODULES OVER A PID, PART II. 389 Let a.x/ D ˛ 1 .x/˛ 2 .x/˛ 3 .x/ . As in the proof of Theorem 8.5.13 , set r 1 .x/ D ˛ 2 .x/˛ 3 .x/ , r 2 .x/ D ˛ 1 .x/˛ 3 .x/ , and r 3 .x/ D ˛ 1 .x/˛ 2 .x/ . For each i , we have to ﬁnd u i .x/ such that u i .x/r i .x/ ± 1 mod ˛ i .x/ . Then we can take y i .x/ to be u i .x/r i .x/ . The results are y 1 .x/ D 11x ² 2 500 r 1 .x/; y 2 .x/ D ² 1 16 r 2 .x/ y 3 .x/ D 81x 2 ² 596x C 1159 2000 r 3 .x/: The Elementary Divisor Decomposition Lemma 8.5.15. Suppose a ﬁnitely generated torsion module M over a principal ideal domain R is an internal direct sum of a collection f C i g of cyclic submodules, each having period a power of a prime. Then for each irreducible p , the sum of those C i that are annihilated by a power of p is equal to MŒpŁ . Proof. Let p 1 ;p 2 ;:::;p s be a list of the irreducibles appearing in an irre- ducible factorization of a period m of M . Denote by AŒp j Ł the sum of those C i that are annihilated by a power of p j . Then AŒp j Ł ³ MŒp j Ł and M is the internal direct product of the
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Unformatted text preview: submodules AŒp j Ł . Since M is also the internal direct product of the sub-modules MŒp j Ł , it follows that AŒp j Ł D MŒp j Ł for all j . n Theorem 8.5.16. (Structure Theorem for Finitely Generated Torsion Mod-ules over a PID, Elementary Divisor Form) Let R be a principal ideal domain, and let M be a (nonzero) ﬁnitely generated torsion module over R . Then M isomorphic to a direct sum of cyclic submodules, each having period a power of an irreducible, M Š M j M i R=.p n i;j j / The number of direct summands, and the annihilator ideals .p n i;j j / of the direct summands are uniquely determined (up to order). Proof. For existence, ﬁrst decompose M as the direct sum of its primary components: M D MŒp 1 Ł ˚ ´´´ ˚ MŒp k Ł...
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