Corollary 1

Let $A$ be a ring, $r$ a nil ideal of $A$ and $h : A \to A/r$ the canonical homomorphism. For every finite orthogonal family $(e'_i)_{i \in I}$ of idempotents of $A/r$ such that $\sum_{i \in I} e'_i = 1$, there exists a finite orthogonal family $(e_i)_{i \in I}$ of idempotents of $A$ such that $\sum_{i \in I} e_i = 1$ and $h(e_i) = e'_i$ for all $i \in I$.

We write $A' = A/r$. We know (Remark following Proposition 13) that

is a homeomorphism, every prime ideal of $A$ containing $r$ by hypothesis. Proposition 15 shows that there exists in $A$ a finite orthogonal family $(e_i)_{i \in I}$, of idempotents such that $\sum_{i \in I} e_i = 1$ and that the image under $ah$ of $V(A'(1 - e'_i))$ is $V(A(1 - e_i))$. But clearly $V(A(1 - e_i))$ is also the image under $ah$ of

as $1 - e'_i$ and $1 - h(e_i)$ are idempotent, Lemma 2 shows that $e'_i = h(e_i)$, whence the corollary.