Lemma 1

Let $a$ be an element of a ring A such that $a-a^2$ is nilpotent. There exists a polynomial P in $X + (X-X^2)\mathbf{Z}[X]$ such that $P(a)$ is idempotent in A.

Let $n$ be a strictly positive integer such that $(a-a^2)^n= 0$. Set P(X) = $1-(1-X^n)^n$. The polynomial P(X) is a multiple of $X^n$, and the polynomial $1-P(X)$ is a multiple of $(1-X)^n$, so $P(X)-P(X)^2$ is a multiple of $(X-X^2)^n$, and we have $P(a) = P(a)^2$. Moreover, $X-P(X)$ is a multiple of X and $1-X$, hence a multiple of $X-X^2$.