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Unformatted text preview: 1 / R=.d s / R n s : If some d i were invertible, then R=.d i / would be the zero module, so could be dropped from the direct sum. But this would display M as generated by fewer than n elements, contradicting the minimality of n . We have proved the existence part of the following fundamental theo-rem: Theorem 8.5.2. (Structure Theorem for Finitely Generated Modules over a PID: Invariant Factor Form) Let R be a principal ideal domain, and let M be a (nonzero) nitely generated module over R . (a) M is a direct sum of cyclic modules, M R=.a 1 / R=.a 2 / R=.a s / R k ; where the a i are nonzero, nonunit elements of R , and a i divides a j for i j ....
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- Fall '08