Fitting lemma
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The Fitting lemma, named after the mathematician Hans Fitting, is a basic statement in abstract algebra. Suppose M is a module over some ring. If M is indecomposable and has finite length, then every endomorphism of M is either bijective or nilpotent.
As an immediate consequence, we see that the endomorphism ring of every finite-length indecomposable module is local.
A version of Fitting's lemma is often used in the representation theory of groups. This is in fact a special case of the version above, since every K-linear representation of a group G can be viewed as a module over the group algebra KG.
To prove Fitting's lemma, we take an endomorphism f of M and consider the following descending sequence of submodules: im(f), im(f 2), im(f 3),... Because M has finite length, this sequence cannot be strictly decreasing forever, so there exists some n with im(f n) = im(f n+1). Then it is not difficult to show that M is the direct sum of im(f n) and ker(f n). Because M is indecomposable, one of those two summands must be equal to M, and the other must be equal to {0}. Depending on which of the two summands is zero, we find that f is bijective or nilpotent.