A note on the separable closedness of a Hensel pair (Q1065878)

We prove that, if (A,\({\mathfrak a})\) is a Hensel pair and f: (A,\({\mathfrak a})\to (B,{\mathfrak b})\) is a strict injective homomorphism of pairs, then A is separably closed in B if and only if every idempotent of B is in A (equivalently if (B,\({\mathfrak b})\) is a strong LI-pair, that is if every idempotent of B/\({\mathfrak b}\) has a unique idempotent lifting in B). As a corollary, we obtain that A is separably closed in \(\hat A,\) the \({\mathfrak a}\)-adic completion of A [this generalizes theorem 1 of the paper by \textit{S. Greco} and \textit{N. Sankaran} in J. Algebra 39, 335-348 (1976; Zbl 0332.13015)].