The Krull-Schmidt Theorem states that a module of finite length, or more generally an object satisfying suitable finiteness conditions, decomposes into a direct sum of indecomposable summands that is unique up to reordering and isomorphism. Named for Wolfgang Krull and Otto Schmidt, it extends the uniqueness of prime factorization from integers to a broad class of algebraic structures, including finite groups and finitely generated modules.
Facts
Partially Attested
Proof YearThe present-day theorem was first proved by Joseph Wedderburn for finite groups in 1909; Robert Remak's 1911 thesis and Otto Schmidt's 1913 paper strengthened and simplified it; Wolfgang Krull extended it to abelian operator groups in 1925, per the History section. StatementThe Krull-Schmidt theorem states that an object satisfying suitable chain conditions, such as a finite group or a module of finite length, decomposes into a finite direct product or direct sum of indecomposable pieces that is unique up to reordering and isomorphism. 1 Classification
Statement Form Connections
Has Statement Form
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.
Proved By
Source Wikipedia: Krull-Schmidt theorem
Sources
1. Wikipedia: Krull-Schmidt theorem
Wikimedia Foundationlead paragraph, first sentence
In mathematics, the Krull-Schmidt theorem states that a group subjected to certain finiteness conditions on chains of subgroups, can be uniquely written as a finite direct product of indecomposable subgroups.
History section, the Wedderburn 1909 sentence
The present-day Krull-Schmidt theorem was first proved by Joseph Wedderburn (Ann. of Math (1909)), for finite groups, though he mentions some credit is due to an earlier study of G.A. Miller where direct products of abelian groups were considered.
Lead section, statement-form reference
In mathematics, the Krull-Schmidt theorem states that a group subjected to certain finiteness conditions on chains of subgroups, can be uniquely written as a finite direct product of indecomposable subgroups.
Proved By: Wolfgang Krull, Lead paragraph
In mathematics, the Krull-Schmidt theorem states that a group subjected to certain finiteness conditions on chains of subgroups, can be uniquely written
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