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: 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. Notes References Module theory Lemmas in algebra