Elementary divisors: Difference between revisions

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In [[algebra]], the '''elementary divisors''' of a [[module (mathematics)|module]] over a [[principal ideal ___domain]] (PID) occur in one form of the [[structure theorem for finitely generated modules over a principal ideal ___domain]].

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 finite sum of the form

::<math>M\cong R^r\oplus \bigoplus_ibigoplus_{i=1}^l R/(q_i) \qquad\text{with }r,l\geq0</math>

:where the <math>(q_i)</math> are proper [[ideal (ring theory)|ideal]]s (<math>(q_i)\neq R</math>) such that <math>(q_1)\supsetsupseteq (q_2)\supsetsupseteq \cdots\supseteq (q_l)</math>, or equivalently <math>q_1\mid q_2\mid\cdots\mid q_l</math>.

The ideals <math>(q_i)</math> are unique; the elements <math>q_i</math> are unique up to [[associatedness]], and are called the ''elementary divisors''. The nonnegative integer <math>r</math> is called the ''free rank'' or ''Betti number'' of the module <math>M</math>.