Is the trace of an idempotent matrix a sum of idempotents?

Question

Let $R$ be a commutative ring, $n$ a positive integer, and $A$ an idempotent $n$ by $n$ matrix with entries in $R$.

Is the trace of $A$ necessarily a sum of idempotents of $R$?

Answer 1

I think a positive answer to the question follows immediately from this MathOverflow answer of Neil Strickland.

EDIT (Aug 8'14). Here are more details. (This almost a copy and paste of Neil Strickland's post.)

Let $x$ and $y$ be indeterminates, and put $$ B(x)=xA+1-A\in R[A,x],\quad f(x)=\det(B(x))\in R[x]. $$ We find that $$ B(xy)=B(x)B(y)\in R[A,x,y],\quad f(xy)=f(x)f(y)\in R[x,y]. $$ We also have $f(1)=1$. It follows that $$ f(x)=\sum_{i=0}^me_i\,x^i $$ for some elements $e_i\in R$ which are orthogonal idempotents, i.e. $$ e_i\,e_j=\delta_{ij}\,e_i,\quad \sum_ie_i=1. $$ We also have $$ \text{tr}(A)=f'(1)=\sum_ii\,e_i. $$