##  [Fitting's Lemma](/fittings-lemma-0) 

 Definition

A lemma in module and linear algebra stating that for an endomorphism f of a finite‑length module M (or a finite-dimensional vector space) there exists N such that M = Ker(f^N) ⊕ Im(f^N); equivalently, M decomposes into a direct sum of a nilpotent part and a part on which f acts invertibly.

 

 

 

 

 

 





## Principle

Principle

Iterating an endomorphism on a module of finite length stabilizes kernels and images; once stabilization occurs the module splits into the generalized nilspace and the generalized unit space, reflecting the dichotomy between nilpotent and automorphism behavior.

 

 

 

 

 





## Demonstration

Demonstration

Concrete example: on a finite-dimensional vector space a linear map has a Jordan decomposition; taking N larger than the maximum size of Jordan blocks with eigenvalue 0 yields the decomposition Ker(f^N) ⊕ Im(f^N).

 

 

 

 

## Misapplication

Misapplication

Applying Fitting's lemma to modules without finite length or to operators on infinite-dimensional spaces without verifying stabilization of kernels and images; assuming the decomposition holds without the finite/stabilization hypothesis can fail.

 

 

 

 

 





## Consequence

Consequence

Provides a canonical splitting used in structure theory (decomposition of modules, primary decomposition, classification of linear operators) and yields control over invariant submodules and the behavior of endomorphisms modulo nilpotent parts.

 

 

 

 

## Reversal

Reversal

Conversely, if a module splits as a direct sum of a nilpotent submodule and an f‑invariant submodule on which f is invertible, then appropriate powers of f realize the decomposition; the lemma and its converse characterize the splitting in finite length contexts.

 

 

 

 

 





## Boundary

Boundary

Requires finite length (or Noetherian/Artinian conditions) to ensure stabilization; it does not automatically apply to arbitrary infinite modules or to operators lacking eventual kernel/image stabilization.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Often compared to Jordan–Chevalley or primary decomposition: Fitting's lemma targets the nilpotent versus unit dichotomy for a single endomorphism on finite‑length modules, whereas Jordan–Chevalley separates semisimple and nilpotent parts in characteristic zero for linear operators.

 

 

 

 

 





## Synthesis

Synthesis

Fitting's lemma isolates the finite‑length phenomenon that repeated application of an endomorphism separates the module into a nilpotent core and an invertible complement, giving a practical and canonical decomposition for module and operator analysis.