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Let $R$ be a commutative ring with $1$. Let $n$ and $k$ be nonnegative integers, and let $A\in\mathrm{M}_n\left(R\right)$ be a matrix such that $A\cdot R^n\cong R^k$ as $R$-modules. Assume that $A^2=\lambda A$ for some $\lambda\in R$. Do we have $\mathrm{Tr}A=\lambda k$ ?

Motivation: This holds for $R$ a field, in both $\lambda=0$ and $\lambda$ invertible cases. But the proofs for these cases are different. I am wondering whether they can be unified - if it works over arbitrary commutative rings, for example, it could.

Oh, and if it holds, it gives a kind of generalization of the Molien series to representations over arbitrary rings, provided the invariant spaces of their symmetric powers are free modules.

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# Pseudo-idempotent matrix generating a free module

Let $R$ be a commutative ring with $1$. Let $n$ and $k$ be nonnegative integers, and let $A\in\mathrm{M}_n\left(R\right)$ be a matrix such that $A\cdot R^n\cong R^k$ as $R$-modules. Assume that $A^2=\lambda A$ for some $\lambda\in R$. Do we have $\mathrm{Tr}A=\lambda k$ ?

Motivation: This holds for $R$ a field, in both $\lambda=0$ and $\lambda$ invertible cases. But the proofs for these cases are different. I am wondering whether they can be unified - if it works over arbitrary commutative rings, for example, it could.