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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- Fall '08
- Algebra, Ring, Abelian group, direct sum, Module theory