Fitting’s lemma

Theorem 1 (Fitting Decomposition Theorem).

Let R be a ring, and M a finite-length module over R. Then for any ϕEnd(M), the endomorphism ringMathworldPlanetmath of M, there is a positive integer n such that


Given ϕEnd(M), it is clear that ker(ϕi)ker(ϕi+1) and im(ϕi)im(ϕi+1) for any positive integer i. Therefore, we have an ascending chain of submodulesMathworldPlanetmath


and a descending chain of submodules


Both chains must be finite, since M has finite length. Therefore, we can find a positive integer n such that

{ker(ϕn)=ker(ϕn+1)=, andim(ϕn)=im(ϕn+1)=.

If uM, then ϕn(u)im(ϕn)=im(ϕ2n). Therefore, ϕn(u)=ϕ2n(v) for some vM. Write u=(u-ϕn(v))+ϕn(v). Applying the ϕn to the first term, we get ϕn(u-ϕn(v))=ϕn(u)-ϕ2n(v)=0, so it is in ker(ϕn). The second term is clearly in im(ϕn). So


Furthermore, if uker(ϕn)im(ϕn), then u=ϕn(v) for some vM. Since ϕ2n(v)=ϕn(u)=0, vker(ϕ2n)=ker(ϕn). Therefore, u=ϕn(v)=0. This shows that we can replace + in the equation above by , proving the theorem. ∎

Stated differently, the theorem says that, given an endomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmath ϕ on M, M can be decomposed into two submodules M1 and M2, such that ϕ restricted to M1 is nilpotentPlanetmathPlanetmathPlanetmath, and ϕ restricted to M2 is an isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath.

A direct consequence of this decomposition property is the famous Fitting Lemma:

Corollary 1 (Fitting Lemma).

In the theorem above, ϕ is either nilpotent (ϕn=0 for some n) or an automorphism iff M is indecomposableMathworldPlanetmath.


Suppose first that M is indecomposable. Then either ker(ϕn)=0 or im(ϕn)=0. If n=1, then the lemma is proved. Suppose n>1. In the former case, any uM is the image of some v under ϕn, so u=ϕ(ϕn-1(v)) and therefore ϕ is onto. If ϕ(u)=0, then ϕn(u)=ϕn-1(ϕ(u))=0, so u=0. This means u is an automorphism. In the latter case, ϕn(u)=0 for any uM, so ϕ is nilpotent.

Now suppose M is not indecomposable. Then writing M=M1M2, where M1 and M2 as proper submodules of M, we can define ϕEnd(M) such that ϕ is the identityPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath on M1 and 0 on M2 (ϕ is a projectionPlanetmathPlanetmath of M onto M1). Since both M1 and M2 are proper, ϕ is neither an automorphism nor nilpotent. ∎

Remark. Another way of stating Fitting Lemma is to say that End(M) is a local ringMathworldPlanetmath iff the finite-length module M is indecomposable. The (unique) maximal idealMathworldPlanetmath in End(M) consists of all nilpotent endomorphisms (and its complement consists of, of course, the automorphisms).

Title Fitting’s lemma
Canonical name FittingsLemma
Date of creation 2013-03-22 17:29:26
Last modified on 2013-03-22 17:29:26
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 9
Author CWoo (3771)
Entry type Theorem
Classification msc 16D10
Classification msc 16S50
Classification msc 13C15
Synonym Fitting lemma
Synonym Fitting decomposition theorem
Defines Fitting’s decomposition theorem