Math 2.0

Associative rings and algebras Modules and ideals Lemma

Fitting's lemma

Builds on Module over a ringSubmoduleLength of a moduleEndomorphism ringIndecomposable module

Statement

Let RR be a ring, let MM be a left RR-module of finite length, and let f∈End⁡R(M)f \in \End_R(M) be an element of its endomorphism ring. Then there is an integer n≥1n \ge 1 such that:

  1. M=im⁡fn⊕ker⁡fnM = \im f^n \oplus \ker f^n, a direct sum of submodules;
  2. the restriction of ff to im⁡fn\im f^n is an automorphism of im⁡fn\im f^n;
  3. the restriction of ff to ker⁡fn\ker f^n is nilpotent.

In particular, if MM is indecomposable, then every endomorphism of MM is either an automorphism or nilpotent.

Remarks

A module of finite length is both Noetherian and Artinian. So the chain ker⁡f⊆ker⁡f2⊆⋯\ker f \subseteq \ker f^2 \subseteq \cdots and the chain im⁡f⊇im⁡f2⊇⋯\im f \supseteq \im f^2 \supseteq \cdots both become constant, say from step nn on. If x∈im⁡fn∩ker⁡fnx \in \im f^n \cap \ker f^n, then x=fn(y)x = f^n(y) with f2n(y)=0f^{2n}(y) = 0, so y∈ker⁡f2n=ker⁡fny \in \ker f^{2n} = \ker f^n and x=0x = 0. For x∈Mx \in M, pick zz with fn(x)=f2n(z)f^n(x) = f^{2n}(z); then x=fn(z)+(x−fn(z))x = f^n(z) + (x - f^n(z)) splits xx into the two parts. For an indecomposable MM the corollary shows that the non-invertible endomorphisms form an ideal, so End⁡R(M)\End_R(M) is a local ring. Finite length is needed: multiplication by 22 on the indecomposable ℤ\Z-module ℤ\Z is neither an automorphism nor nilpotent.

Going deeper

For a student

Fitting's lemma is the module version of the splitting of a linear map of a finite-dimensional space into an invertible part and a nilpotent part. Its main use is the corollary that the endomorphism ring of an indecomposable module of finite length is local. This locality is exactly what the proof of the Krull–Schmidt theorem needs.

Reading path

To understand this page from the ground up, read these 19 pages in order. The foundations come first.

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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 lemmastatement
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

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Fitting_lemma
About this page
TagF52B2JMEHY
URIm2://F52B2JMEHY
Kindstatement, lemma
WritingW0 stub
MSC16D70, 16S50
StatusThis copy of the site has no build data. Run m2 status F52B2JMEHY.
AboutEndomorphism ring
UsesModule over a ring, Submodule, Length of a module, Endomorphism ring, Indecomposable module
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