Math 2.0

Associative rings and algebras Modules and ideals Theorem

Krull–Schmidt theorem

Builds on Module over a ringLength of a moduleIndecomposable moduleFitting's lemma

Statement

Let RR be a ring and let MM be a nonzero left RR-module of finite length.

  1. MM is a direct sum M=M1⊕⋯⊕MmM = M_1 \oplus \cdots \oplus M_m of finitely many indecomposable submodules.
  2. If also M=N1⊕⋯⊕NnM = N_1 \oplus \cdots \oplus N_n with each NjN_j indecomposable, then m=nm = n and there is a permutation σ\sigma of {1,…,m}\set{1, \dots, m} with Mi≅Nσ(i)M_i \cong N_{\sigma(i)} for all ii.

Remarks

Existence follows by induction on the length: a module that is not indecomposable splits into two summands of smaller length. For uniqueness, Fitting's lemma shows that the endomorphism ring of each indecomposable summand is local. An exchange argument then matches M1M_1 with some NjN_j and replaces one by the other, and induction on mm finishes. Example: a finite abelian group is a ℤ\Z-module of finite length, and its decomposition into cyclic groups of prime power order is unique up to order and isomorphism. Uniqueness can fail without finite length: in the ring of integers 𝒪\mathcal{O} of a number field with nontrivial class group, a non-principal ideal II satisfies I⊕I−1≅𝒪⊕𝒪I \oplus I^{-1} \cong \mathcal{O} \oplus \mathcal{O}, though I≇𝒪I \not\cong \mathcal{O}.

Going deeper

For a student

The theorem reduces the study of modules of finite length to that of indecomposable modules. For finite-dimensional algebras over a field, every finite-dimensional module has finite length, so the theorem applies; this is the base of representation theory of quivers and algebras. The same proof works in any additive category in which idempotents split and endomorphism rings of indecomposable objects are local.

Reading path

To understand this page from the ground up, read these 20 pages in order. The foundations come first.

Show the path as a graph
Endomorphism ring, concept, no evidence levelEndomorphism ringconceptGroup, concept, no evidence levelGroupconceptIndecomposable module, concept, no evidence levelIndecomposable moduleconceptModule over a ring, definition, no evidence levelModule over a ringdefinitionConjugate elements, definition, no evidence levelConjugate elementsdefinitionRing, definition, no evidence levelRingdefinitionSubgroup, definition, no evidence levelSubgroupdefinitionSubmodule, definition, no evidence levelSubmoduledefinitionConjugate subgroup, definition, no evidence levelConjugate subgroupdefinitionLeft coset, definition, no evidence levelLeft cosetdefinitionRight coset, definition, no evidence levelRight cosetdefinitionNormal subgroup, definition, no evidence levelNormal subgroupdefinitionSimple group, definition, no evidence levelSimple groupdefinitionNormality via left and right cosets, statement, no evidence levelNormality via left an…statementQuotient group, definition, no evidence levelQuotient groupdefinitionSubnormal series, definition, no evidence levelSubnormal seriesdefinitionComposition series, definition, no evidence levelComposition seriesdefinitionLength of a module, definition, no evidence levelLength of a moduledefinitionFitting's lemma, statement, no evidence levelFitting's lemmastatementKrull–Schmidt theorem, statement, no evidence levelKrull–Schmidt theoremstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Endomorphism ring concept
  2. Group concept
  3. Indecomposable module concept
  4. Module over a ring definition
  5. Conjugate elements definition
  6. Ring definition
  7. Subgroup definition
  8. Submodule definition
  9. Conjugate subgroup definition
  10. Left coset definition
  11. Right coset definition
  12. Normal subgroup definition
  13. Simple group definition
  14. Normality via left and right cosets statement
  15. Quotient group definition
  16. Subnormal series definition
  17. Composition series definition
  18. Length of a module definition
  19. Fitting's lemma statement
  20. Krull–Schmidt theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Krull%E2%80%93Schmidt_theorem
About this page
Tag94M0BPV1W4
URIm2://94M0BPV1W4
Kindstatement, theorem
WritingW0 stub
MSC16D70
StatusThis copy of the site has no build data. Run m2 status 94M0BPV1W4.
AboutIndecomposable module
UsesModule over a ring, Length of a module, Indecomposable module, Fitting's lemma
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