Fitting lemma

To prove Fitting's lemma, we take an endomorphism f of M and consider the following two chains of submodules:

• The first is the descending chain ${\displaystyle \mathrm {im} (f)\supseteq \mathrm {im} (f^{2})\supseteq \mathrm {im} (f^{3})\supseteq \ldots }$,
• the second is the ascending chain ${\displaystyle \mathrm {ker} (f)\subseteq \mathrm {ker} (f^{2})\subseteq \mathrm {ker} (f^{3})\subseteq \ldots }$

Because ${\displaystyle M}$ has finite length, both of these chains must eventually stabilize, so there is some ${\displaystyle n}$ with ${\displaystyle \mathrm {im} (f^{n})=\mathrm {im} (f^{n'})}$ for all ${\displaystyle n'\geq n}$, and some ${\displaystyle m}$ with ${\displaystyle \mathrm {ker} (f^{m})=\mathrm {ker} (f^{m'})}$ for all ${\displaystyle m'\geq m.}$

Let now ${\displaystyle k=\max\{n,m\}}$, and note that by construction ${\displaystyle \mathrm {im} (f^{2k})=\mathrm {im} (f^{k})}$ and ${\displaystyle \mathrm {ker} (f^{2k})=\mathrm {ker} (f^{k}).}$

We claim that ${\displaystyle \mathrm {ker} (f^{k})\cap \mathrm {im} (f^{k})=0}$. Indeed, every ${\displaystyle x\in \mathrm {ker} (f^{k})\cap \mathrm {im} (f^{k})}$ satisfies ${\displaystyle x=f^{k}(y)}$ for some ${\displaystyle y\in M}$ but also ${\displaystyle f^{k}(x)=0}$, so that ${\displaystyle 0=f^{k}(x)=f^{k}(f^{k}(y))=f^{2k}(y)}$, therefore ${\displaystyle y\in \mathrm {ker} (f^{2k})=\mathrm {ker} (f^{k})}$ and thus ${\displaystyle x=f^{k}(y)=0.}$

Moreover, ${\displaystyle \mathrm {ker} (f^{k})+\mathrm {im} (f^{k})=M}$: for every ${\displaystyle x\in M}$, there exists some ${\displaystyle y\in M}$ such that ${\displaystyle f^{k}(x)=f^{2k}(y)}$ (since ${\displaystyle f^{k}(x)\in \mathrm {im} (f^{k})=\mathrm {im} (f^{2k})}$), and thus ${\displaystyle f^{k}(x-f^{k}(y))=f^{k}(x)-f^{2k}(y)=0}$, so that ${\displaystyle x-f^{k}(y)\in \mathrm {ker} (f^{k})}$ and thus ${\displaystyle x\in \mathrm {ker} (f^{k})+f^{k}(y)\subseteq \mathrm {ker} (f^{k})+\mathrm {im} (f^{k}).}$

Consequently, ${\displaystyle M}$ is the direct sum of ${\displaystyle \mathrm {im} (f^{k})}$ and ${\displaystyle \mathrm {ker} (f^{k})}$. (This statement is also known as the Fitting decomposition theorem.) Because ${\displaystyle M}$ is indecomposable, one of those two summands must be equal to ${\displaystyle M}$ and the other must be the zero submodule. Depending on which of the two summands is zero, we find that ${\displaystyle f}$ is either bijective or nilpotent.^[2]