Bound on nilpotency index of endomorphisms 
Let $A$ be a Noetherian ring (commutative with $1$) and $M$ a finitely generated $A$-module. I want to show that there exists a bound $n$ such that for every nilpotent endomorphism $T : M \to M$ we have $T^n = 0$.

There are two examples in mind:


*

*$M$ a module over a field $K$, that is $M = K^n$ a finite-dimensional vector space. In that case, $n$ is the required upper bound.

*$A=K[x]/(x^d)$ as a module over itself. In that case the dimension is $1$ (a basis is $\{ 1 \}$) but the upper bound is $d$ (consider the endomorphism $y \mapsto x y$).


This is not a complete answer, but covers both examples where the modules have finite length ($n$, respectively $d$). 
Consider $M$ of finite length $n$. Then, by Fitting's Lemma, for every $T\in\operatorname{End}_A(M)$ we have $M=\operatorname{Im}T^n\oplus\ker T^n$. If $T$ is nilpotent, then there exists $m\ge 1$ such that $T^m=0$. If $m\le n$, then $T^n=0$. If $m>n$, then we also have $M=\operatorname{Im}T^m\oplus\ker T^m$. Since $T^m=0$ it follows that $M=\ker T^m=\ker T^n$, thus $\operatorname{Im}T^n=0$, i.e. $T^n=0$.
Remark. The proof of Fitting's Lemma goes like this: Assume that $\ker T\neq 0$. Then $\ker T\subseteq\ker T^2\subseteq\cdots$. Suppose the following inclusions $\ker T\subset \cdots\subset\ker T^i$ are strict, and $\ker T^i=\ker T^{i+1}$. Now we can deduce two things: the length of $\ker T^i$ is at least $i$, so $i\le n$, and $\ker T^{i+1}=\ker T^{i+2}=\cdots$. On the other side, $\operatorname{Im}T\supseteq\operatorname{Im}T^2\supseteq\cdots$. Suppose the following inclusions $\operatorname{Im}T\supset\cdots\supset\operatorname{Im}T^j$ are strict, and $\operatorname{Im}T^j=\operatorname{Im}T^{j+1}$. As before we deduce that $j\le n$ and $\operatorname{Im}T^j=\operatorname{Im}T^{j+1}=\cdots$. Now for any $k\ge\max(i,j)$ we get $M=\operatorname{Im}T^k\oplus\ker T^k$. 
(I've added this proof since the usual proofs of Fitting's Lemma do not say it explicitely that the equality holds for any $k\ge n$, where $n$ is the length of $M$.)
