PlanetMath: Fitting's lemma

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Fitting's lemma

(Theorem)

Theorem 1 (Fitting Decomposition Theorem) Let be a ring, and a finite-length module over . Then for any $\phi \in \operatorname{End}(M)$ , the endomorphism ring of , there is a positive integer 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

. 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

has finite length. Therefore, we can find a positive integer

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 , can be decomposed into two submodules and , such that $\phi$ restricted to is nilpotent, and $\phi$ restricted to 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 ) or an automorphism iff is indecomposable.

Proof. Suppose first that

is indecomposable. Then either $\ker(\phi^n)=0$ or $\operatorname{im}(\phi^n)=0$ . If

, then the lemma is proved. Suppose

. In the former case, any $u\in M$ is the image of some

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

. This means

is an automorphism. In the latter case, $\phi^n(u)=0$ for any $u\in M$ , so $\phi$ is nilpotent.

Now suppose is not indecomposable. Then writing $M=M_1\oplus M_2$ , where and as proper submodules of , we can define $\phi\in \operatorname{End}(M)$ such that $\phi$ is the identity on and 0 on ( $\phi$ is a projection of onto ). Since both and 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 is indecomposable. The (unique) maximal ideal in $\operatorname{End}(M)$ consists of all nilpotent endomorphisms (and its complement consists of, of course, the automorphisms).

"Fitting's lemma" is owned by CWoo.

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Other names:

Fitting lemma, Fitting decomposition theorem

Also defines:

Fitting's decomposition theorem

This object's parent.

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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 MSC:	13C15 (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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