Consider the following Condition (C) on a positive integer $k$:

(C) If $R$ is a commutative ring, if $F$ is a free $R$-module, and if $f$ is an endomorphism of $F$, then $f$ is an $R$-linear combination of $k$ idempotent endomorphisms of $F$.

The question is

> (a) Is there a positive integer $k$ satisfying Condition (C)?

If $R$ is a field and $F$ finite dimensional, [Clément de Seguins Pazzis]( https://mathoverflow.net/users/34951/clement-de-seguins-pazzis) proves in [this paper](http://arxiv.org/abs/0907.4949) that $f$ is an $R$-linear combination of three idempotents. [George Lowther](https://mathoverflow.net/users/1004/george-lowther) announced a generalization of this result to the infinite dimensional case. 

In [this post](https://math.stackexchange.com/a/887623/660) George Lowther shows the following:

> (b) Any endomorphism of any free $R$-module of *infinite rank* is a $\mathbb Z$-linear combination of twelve idempotents. (The argument works even if $R$ is not commutative.)

For any positive integer $n$ let $R_n$ be the ring 
$$
R_n:=\mathbb Z[(a_{ij})_{1\le i,j\le n}],
$$
where the $a_{ij}$ are indeterminates, and form the $n$ by $n$ matrix with entries in $R_n$
$$
A_n:=(a_{ij})_{1\le i,j\le n}.
$$ 
In view of (b), Question (a) is equivalent to

> Is there a positive integer $k$ such that, for all $n$, the matrix $A_n$ is an $R_n$-linear combination of $k$ idempotents of $M_n(R_n)$? 

I asked a [related question](https://math.stackexchange.com/q/891993/660) on Mathematics Stack Exchange.