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("f" "n") = im("f" "n"+1). Likewise (as "M" has finite length) the second sequence cannot be "strictly" increasing forever, so there exists some "m" with ker("f" "m") = ker("f" "m"+1). It is easily seen that im("f" "n") = im("f" "n"+1) yields im("f" "n") = im("f" "n"+1) = im("f" "n"+2) = …, and that ker("f" "m") = ker("f" "m"+1) yields ker("f" "m") = ker("f" "m"+1) = ker("f" "m"+2) = …. Putting "k" = max("m","n"), it now follows that im("f" "k") = im("f" 2"k") and ker("f" "k") = ker("f" 2"k"). Hence, formula_1 (because every formula_2 satisfies formula_3 for some formula_4 but also formula_5, so that formula_6, therefore formula_7 and thus formula_8) and formula_9 (since for every formula_10, there exists some formula_4 such that formula_12 (since formula_13), and thus formula_14, so that formula_15 and thus formula_16). Consequently, "M" is the direct sum of im("f" "k") and ker("f" "k"). 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.