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 algebraKG.
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 ,
the second sequence is the ascending sequence
Because has finite length, both of these sequences must eventually stabilize, so there is some with for all , and some with for all .
Let now , and note that by construction and .
We claim that . Indeed, every satisfies for some but also , so that , therefore and thus .
Moreover, : for every , there exists some such that (since ), and thus , so that and thus .
Consequently, is the direct sum of and . (This statement is also known as the Fitting decomposition theorem.) Because is indecomposable, one of those two summands must be equal to , and the other must be the trivial submodule. Depending on which of the two summands is zero, we find that is either bijective or nilpotent.
^Jacobson, A lemma before Theorem 3.7. harvnb error: no target: CITEREFJacobson (help)
^"Fitting's lemma". PlanetMath. Retrieved 22 July 2022.