Fitting’s lemma
Theorem 1 (Fitting Decomposition Theorem).
Let R be a ring, and M a finite-length module over R. Then for any ϕ∈End(M), the endomorphism ring of M, there is a positive integer n such that
M=ker(ϕn)⊕im(ϕn). |
Proof.
Given ϕ∈End(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 submodules
ker(ϕ)⊆⋯⊆ker(ϕi)⊆ker(ϕi+1)⊆⋯, |
and a descending chain of submodules
im(ϕ)⊇⋯⊇im(ϕi)⊇im(ϕi+1)⊇⋯. |
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 u∈M, then ϕn(u)∈im(ϕn)=im(ϕ2n). Therefore, ϕn(u)=ϕ2n(v) for some v∈M. 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
M=ker(ϕn)+im(ϕn). |
Furthermore, if u∈ker(ϕn)∩im(ϕn), then u=ϕn(v) for some v∈M. Since ϕ2n(v)=ϕn(u)=0, v∈ker(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 endomorphism ϕ on M, M can be decomposed into two submodules M1 and M2, such that ϕ restricted to M1 is nilpotent
, and ϕ restricted to M2 is an isomorphism
.
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 indecomposable.
Proof.
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 u∈M 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 u∈M, so ϕ is nilpotent.
Now suppose M is not indecomposable. Then writing M=M1⊕M2, where M1 and M2 as proper submodules of M, we can define ϕ∈End(M) such that ϕ is the identity on M1 and 0 on M2 (ϕ is a projection
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 ring iff the finite-length module M is indecomposable. The (unique) maximal ideal
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 |