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$q_i\neq 1$ should be $q_i$ not invertible, but is anyway implied by primary ideal |
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If <math>R</math> is a [[Principal ideal ___domain|PID]] and <math>M</math> a finitely generated <math>R</math>-module, then ''M'' is isomorphic to a unique sum of the form
::<math>M\cong R^r\oplus \bigoplus_i R/(q_i)</math>
:where
The ideals <math>(q_i)</math> are unique (up to order); the elements <math>q_i</math> are unique up to [[associatedness]], and are called the ''elementary divisors''. Note that in a PID, primary ideals are powers of primes, so the elementary divisors <math>(q_i)=(p_i^{r_i}) = (p_i)^{r_i}</math>. The nonnegative integer <math>r</math> is called the ''free rank'' or ''Betti number'' of the module <math>M</math>.
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