Every finitely generated module over a PID is a direct sum of a free module and cyclic prime-power torsion modules.
For a finitely generated module over a principal ideal domain, its invariant factors are nonzero nonunits in a decomposition , ordered by . They are unique up to multiplication by units and describe the torsion submodule. Over , making act as a linear operator recovers the invariant factors of a linear operator used in its rational canonical form.
Every finitely generated torsion-free module over a principal ideal domain is free. In the structure theorem, torsion-freeness removes every cyclic torsion summand and leaves only the free summand.
An integral matrix satisfying makes a module over the Eisenstein integers by letting the primitive cube root act as . This module is finitely generated and torsion-free, hence free. Since the Eisenstein integers have integer rank two, such matrices exist only for ; for that dimension every one is integrally conjugate to
An elementary divisor is a prime power occurring in the cyclic decomposition of a finitely generated torsion module.
A finite abelian group is indecomposable exactly when it is cyclic of prime-power order. The structure theorem proves necessity. Conversely, every nonzero subgroup of contains its unique subgroup of order , so two nonzero subgroups cannot be complementary direct summands.
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The Structure Theorem for finitely generated modules over a principal ideal domain (PID) is a fundamental result in abstract algebra, specifically in the study of modules over rings. It describes the classification of finitely generated modules over a PID in terms of simpler components. Here’s a concise statement of the theorem: Let \( R \) be a principal ideal domain, and let \( M \) be a finitely generated \( R \)-module.