Let $A$ be a commutative ring with unit and let $P$ be a projective $A$-module finitely generated. By definition, there exists an $A$-module $P'$ such that $P\oplus P'$ is free of finite rank $r$. If $f$ is an endomorphism of $P$, its trace $\operatorname{Tr}(f|P)$ is defined to be $\operatorname{Tr}(f\oplus 0|P\oplus P')$. We can show that this definition does not depend on the choice of $P'$.

Let $Q$ be a submodule of $P$ stable by $f$ which is projective and finitely generated. Assume that there is a positive integer $n$ such that $f^n(P)\subset Q$. If $P/Q$ would have been projective and finitely generated, I would have conclude that $\operatorname{Tr}(f|P)=\operatorname{Tr}(f|Q)+s$, where $s$ is a nilpotent element of $A$, as $f$ acts nilpotently on $P/Q$. But, does this still holds in general?

Any idea is welcome. Many thanks!

(EDIT: As Darij Grinberg pointed me out in comments, the trace of a nilpotent endomorphism of a projective finitely generated module is a sum of nilpotents elements of $A$. As $A$ is commutative, the trace is again nilpotent and not necessarily zero)

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