A characterization of semilocal inertial coefficient rings

A commutative ring $R$ with identity is called an inertial coefficient ring if every finitely generated $R$-algebra $A$ with $A/N$ separable over $R$ contains a separable subalgebra $S$ such that $S + N = A$, where $N$ is the Jacobson radical of $A$. Thus inertial coefficient rings are those commutative rings $R$ for which a generalization of the Wedderburn Principal Theorem holds for suitable $R$-algebras. Our purpose is to prove that a commutative ring with only finitely many maximal ideals is an inertial coefficient ring (if and) only if it is a finite direct sum of Hensel rings.