Lemma 2

Let $\mathbf{A}$ be a ring and $m$ an ideal of $\mathbf{A}$ such that the ordered pair $(\mathbf{A}, m)$ satisfies Hensel’s conditions. Let $\mathbf{B}$ be the quotient ring $\mathbf{A}/m$ and $\pi : \mathbf{A} \to \mathbf{B}$ be the canonical homomorphism. For every idempotent $c$ of $\mathbf{B}$ there exists a unique idempotent $e$ of $\mathbf{A}$ such that $\pi(e) = c$.

Let $a$ be such that $\pi(a) = c$; Corollary 1 to Theorem 2 of no. 5 may be applied to the polynomial $f(X) = X^2 - X$ in $\mathbf{A}[X]$ and the element $a \in \mathbf{A}$. Then $f'(a) = 2a - 1$ and, as $\pi(2a - 1) = 2c - 1$ and $(2c - 1)^2 = 1$ in $\mathbf{B}$, $2c - 1$ is invertible in $\mathbf{B}$ and hence $2a - 1$ is invertible in $\mathbf{A}$ (§ 2, no. 13, Lemma 3). As $f(a) \in m$, Corollary 1 to Theorem 2 of no. 5 immediately gives the existence and uniqueness of $e$.