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Unformatted text preview: submodules Ap j . Since M is also the internal direct product of the sub-modules Mp j , it follows that Ap j D Mp j 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) nitely generated torsion module over R . Then M isomorphic to a direct sum of cyclic submodules, each having period a power of an irreducible, M 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, rst decompose M as the direct sum of its primary components: M D Mp 1 Mp k...
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
- Fall '08