Unformatted text preview: 1 / ˚ ²²² ˚ 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 theorem: 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) ﬁnitely generated module over R . (a) M is a direct sum of cyclic modules, M Š R=.a 1 / ˚ R=.a 2 / ˚ ²²² ˚ 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
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 Algebra

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