Fitting lemma


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 an automorphism 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.

Proof

To prove Fitting's lemma, we take an endomorphism f of M and consider the following two sequences of submodules:
Because M has finite length, the first sequence cannot be strictly decreasing forever, so there exists some n with im = im. Likewise the second sequence cannot be strictly increasing forever, so there exists some m with ker = ker. It is easily seen that im = im yields im = im = im = …, and that ker = ker yields ker = ker = ker = …. Putting k = max, it now follows that im = im and ker = ker. Hence, and . Consequently, M is the direct sum of im and ker. Because M is indecomposable, one of those two summands must be equal to M, and the other must be equal to. Depending on which of the two summands is zero, we find that f is bijective or nilpotent.