The theorem is that if we have a finite length module M (Noetherian and Artinian), and a map f is endomorphism. Then, we can decompose M = Ker($f^n$) $\oplus$ Im($f^n$).
I understand the proof is that we use the idea: finite length property of M makes ascending chain of Ker($f^i$) stabilized, and the descending chain of Im($f^i$) stabilized.
I got a stupid question, why not use the first isomorphism thm directly, since f is R-linear, a module homomorphism, so M/Ker($f^n$)=Im($f^n$), so we can get M = Ker($f^n$)$\oplus$Im($f^n$). I know there must be some thing wrong......please help.
Hint: Suppose $f$ is nilpotent $f^2=0, f\neq 0, Im(f)\subset Ker f$, so the result is not true for every n,but for $n$ for which $Ker f^{n+1}=Ker f^n$ and $Im f^{n+1}=Imf^n$ which existence is a consequence of the hypothesis.