Every finitely generated module over a principal ideal domain decomposes uniquely as a direct sum of a free module and cyclic torsion modules. It generalizes the classification of finitely generated abelian groups and underlies the theory of canonical matrix forms such as Jordan normal form.
Facts
StatementEvery finitely generated module over a principal ideal domain is isomorphic to a direct sum of a free module of finite rank and finitely many cyclic torsion modules; the theorem has two standard forms, the invariant factor decomposition, where each torsion summand's order divides the next, and the primary decomposition, where each torsion summand has order a power of a single prime. 1 Classification
Statement Form Connections
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Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
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Sources
1. Structure theorem for finitely generated modules over a principal ideal domain (Wikipedia)
Wikimedia FoundationIntroductionQuote, Introduction
In mathematics, in the field of abstract algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that finitely generated modules over a principal ideal domain (PID) can be uniquely decomposed in much the same way that integers have a prime factorization.
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