Proposition 7

Let $\mathfrak{a}$ be a two-sided nil ideal of A, and let $\overline{e}$ be an idempotent in the ring $A/\mathfrak{a}$. There exists an idempotent $e$ in A whose canonical image in $A/\mathfrak{a}$ is equal to $\overline{e}$.

Let $a$ be an arbitrary representative of $\overline{e}$ in A. The element $a-a^2$ of A is nilpotent because it belongs to $\mathfrak{a}$. Choose a polynomial $P\in \mathbf{Z}[X]$ satisfying the conditions of Lemma 1. We have $a-P(a)\in A(a-a^2)$, so the element $e= P(a)$ of A has the desired property.

Suppose that $\overline{e}$ belongs to the center of the ring $A/\mathfrak{a}$. There does not necessarily exist an idempotent $e$ in the center Z of A lifting $\overline{e}$ (VIII, p. 172, Exercise 31). However, if $\overline{e}$ belongs to the image of Z in $A/\mathfrak{a}$, then it lifts to an idempotent in Z because $Z\cap \mathfrak{a}$ is a nil ideal of Z.