Lemma 15.11.9. Let $I \subset J \subset A$ be ideals of a ring $A$. The following are equivalent

$(A, I)$ and $(A/I, J/I)$ are henselian pairs, and

$(A, J)$ is an henselian pair.

Lemma 15.11.9. Let $I \subset J \subset A$ be ideals of a ring $A$. The following are equivalent

$(A, I)$ and $(A/I, J/I)$ are henselian pairs, and

$(A, J)$ is an henselian pair.

**Proof.**
Assume (1). Let $B$ be an integral $A$-algebra. Consider the ring maps

\[ B \to B/IB \to B/JB \]

By Lemma 15.11.6 we find that both arrows induce bijections on idempotents. Hence so does the composition. Whence $(A, J)$ is a henselian pair by Lemma 15.11.6.

Conversely, assume (2) holds. Then $(A/I, J/I)$ is a henselian pair by Lemma 15.11.8. Let $B$ be an integral $A$-algebra. Consider the ring maps

\[ B \to B/IB \to B/JB \]

By Lemma 15.11.6 we find that the composition and the second arrow induce bijections on idempotents. Hence so does the first arrow. It follows that $(A, I)$ is a henselian pair (by the lemma again). $\square$

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