- Fitting lemma
The Fitting lemma, named after the mathematician
Hans Fitting , is a basic statement inabstract algebra . Suppose "M" is a module over some ring. If "M" is indecomposable and has finite length, then everyendomorphism of "M" is eitherbijective ornilpotent .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 two sequences of submodules. The first sequence is the descending sequence im("f"), im("f" 2), im("f" 3),..., the second sequence is the ascending sequence ker("f"), ker("f" 2), ker("f" 3),.... 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). Putting "k" = max("m","n" ), it is not difficult to show that "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.
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