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[parent] Fitting's lemma (Theorem)
Theorem 1 (Fitting Decomposition Theorem)   Let $ R$ be a ring, and $ M$ a finite-length module over $ R$. Then for any $ \phi \in \operatorname{End}(M)$, the endomorphism ring of $ M$, there is a positive integer $ n$ such that
$\displaystyle M=\ker(\phi^n)\oplus \operatorname{im}(\phi^n).$
Proof. Given $ \phi\in \operatorname{End}(M)$, it is clear that $ \ker(\phi^i)\subseteq \ker(\phi^{i+1})$ and $ \operatorname{im}(\phi^i)\supseteq \operatorname{im}(\phi^{i+1})$ for any positive integer $ i$. Therefore, we have an ascending chain of submodules
$\displaystyle \ker(\phi)\subseteq \cdots \subseteq \ker(\phi^i)\subseteq \ker(\phi^{i+1}) \subseteq \cdots,$
and a descending chain of submodules
$\displaystyle \operatorname{im}(\phi)\supseteq \cdots \supseteq \operatorname{im}(\phi^i)\supseteq \operatorname{im}(\phi^{i+1}) \supseteq \cdots.$
Both chains must be finite, since $ M$ has finite length. Therefore, we can find a positive integer $ n$ such that
\begin{displaymath} \left\{ \begin{array}{l} \ker(\phi^n)=\ker(\phi^{n+1})=\cdot... ...^n)= \operatorname{im}(\phi^{n+1}) =\cdots. \end{array}\right. \end{displaymath}
If $ u\in M$, then $ \phi^n(u)\in \operatorname{im}(\phi^n)=\operatorname{im}(\phi^{2n})$. Therefore, $ \phi^n(u)=\phi^{2n}(v)$ for some $ v\in M$. Write $ u=(u-\phi^n(v))+\phi^n(v)$. Applying the $ \phi^n$ to the first term, we get $ \phi^n(u-\phi^n(v))=\phi^n(u)-\phi^{2n}(v)=0$, so it is in $ \ker(\phi^n)$. The second term is clearly in $ \operatorname{im}(\phi^n)$. So
$\displaystyle M=\ker(\phi^n)+\operatorname{im}(\phi^n).$
Furthermore, if $ u\in \ker(\phi^n)\cap \operatorname{im}(\phi^n)$, then $ u=\phi^n(v)$ for some $ v\in M$. Since $ \phi^{2n}(v)=\phi^n(u)=0$, $ v\in \ker(\phi^{2n})=\ker(\phi^n)$. Therefore, $ u=\phi^n(v)=0$. This shows that we can replace $ +$ in the equation above by $ \oplus$, proving the theorem. $ \qedsymbol$

Stated differently, the theorem says that, given an endomorphism $ \phi$ on $ M$, $ M$ can be decomposed into two submodules $ M_1$ and $ M_2$, such that $ \phi$ restricted to $ M_1$ is nilpotent, and $ \phi$ restricted to $ M_2$ is an isomorphism.

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

Corollary 1 (Fitting Lemma)   In the theorem above, $ \phi$ is either nilpotent ($ \phi^n=0$ for some $ n$) or an automorphism iff $ M$ is indecomposable.
Proof. Suppose first that $ M$ is indecomposable. Then either $ \ker(\phi^n)=0$ or $ \operatorname{im}(\phi^n)=0$. If $ n=1$, then the lemma is proved. Suppose $ n>1$. In the former case, any $ u\in M$ is the image of some $ v$ under $ \phi^n$, so $ u=\phi(\phi^{n-1}(v))$ and therefore $ \phi$ is onto. If $ \phi(u)=0$, then $ \phi^n(u)=\phi^{n-1}(\phi(u))=0$, so $ u=0$. This means $ u$ is an automorphism. In the latter case, $ \phi^n(u)=0$ for any $ u\in M$, so $ \phi$ is nilpotent.

Now suppose $ M$ is not indecomposable. Then writing $ M=M_1\oplus M_2$, where $ M_1$ and $ M_2$ as proper submodules of $ M$, we can define $ \phi\in \operatorname{End}(M)$ such that $ \phi$ is the identity on $ M_1$ and 0 on $ M_2$ ($ \phi$ is a projection of $ M$ onto $ M_1$). Since both $ M_1$ and $ M_2$ are proper, $ \phi$ is neither an automorphism nor nilpotent. $ \qedsymbol$

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



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Other names:  Fitting lemma, Fitting decomposition theorem
Also defines:  Fitting's decomposition theorem

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Cross-references: complement, maximal ideal, local ring, projection, identity, onto, image, indecomposable, iff, property, consequence, nilpotent, restricted, endomorphism, equation, term, finite length, finite, submodules, chain, clear, integer, positive, endomorphism ring, finite-length module, ring
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This is version 6 of Fitting's lemma, born on 2007-08-20, modified 2008-04-30.
Object id is 9878, canonical name is FittingsLemma.
Accessed 1712 times total.

Classification:
AMS MSC13C15 (Commutative rings and algebras :: Theory of modules and ideals :: Dimension theory, depth, related rings )
 16D10 (Associative rings and algebras :: Modules, bimodules and ideals :: General module theory)
 16S50 (Associative rings and algebras :: Rings and algebras arising under various constructions :: Endomorphism rings; matrix rings)

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