For an endomorphism of a finite-length module , sufficiently large givesIf is indecomposable, every endomorphism is therefore either invertible or nilpotent, so its endomorphism ring is local.
Every linear endomorphism of a finite-dimensional vector space admits the displayed direct sum of vector spaces. It separates the part on which repeated application is invertible from the part eventually sent to zero.
Proof. Stabilization of images of powers of a linear map gives . The rank-nullity theorem and nested null spaces give . If also satisfies , then , so . Thus the displayed summands have zero intersection. Their dimensions sum to by the rank-nullity theorem, proving the decomposition.
Both summands are preserved by . On , stabilization makes a surjective linear map, hence invertible in finite dimension. On , its th power is zero, so the restriction is a nilpotent linear map.
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The Fitting lemma, often mentioned in the context of group theory and representation theory, primarily deals with nilpotent groups and their substructures. It provides insight into the relationship between normal subgroups and the structure of groups. Here’s a basic overview of the Fitting lemma: ### Fitting Lemma 1.