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

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

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

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

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

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

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

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

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.