Lemma 10.32.7. Let $A$ be a possibly noncommutative algebra. Let $e \in A$ be an element such that $x = e^2 - e$ is nilpotent. Then there exists an idempotent of the form $e' = e + x(\sum a_{i, j}e^ ix^ j) \in A$ with $a_{i, j} \in \mathbf{Z}$.

Proof. Consider the ring $R_ n = \mathbf{Z}[e]/((e^2 - e)^ n)$. It is clear that if we can prove the result for each $R_ n$ then the lemma follows. In $R_ n$ consider the ideal $I = (e^2 - e)$ and apply Lemma 10.32.6. $\square$

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