Fitting lemma

For an endomorphism of a finite-length module, sufficiently high powers split the module as the direct sum of the kernel and image of that power.

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Ascending chain condition: A condition that every ascending chain of subobjects eventually stabilizes. The kernels of successive powers stabilize because a finite-length module satisfies this condition.
Module (mathematics): An abelian group equipped with scalar multiplication by elements of a ring. The lemma applies to modules of finite length over a ring.
Fitting decomposition: The decomposition of a finite-length module into stable kernel and image parts for an endomorphism. It names the structural splitting furnished by the lemma.
Cayley–Hamilton theorem: The theorem that every square matrix satisfies its characteristic polynomial. It yields polynomial relations for operators, while Fitting's lemma relies on stabilization of submodules.
Descending chain condition: A condition that every descending chain of subobjects eventually stabilizes. The images of successive powers stabilize for this reason.
Endomorphism: A morphism from a mathematical object to itself. The lemma starts with a module endomorphism and studies its iterated powers.
Jordan–Chevalley decomposition: A decomposition of a suitable linear operator into commuting semisimple and nilpotent parts. Fitting's splitting helps isolate nilpotent behavior in finite-dimensional linear algebra.
Primary decomposition theorem: A decomposition of a module into components associated with distinct factors of an annihilating polynomial. It extends the kernel-image split from the zero-root component to several polynomial factors.
Kernel (algebra): The submodule mapped to zero by a module homomorphism. The stable kernel supplies the part on which a high power vanishes.
Finite-length module: A module with a finite composition series. Finite length ensures the chains of kernels and images eventually stabilize.