\(H\)-separable rings and their Hopf-Galois extensions (Q1266286)

The authors show that for a ring extension \(A/B\) with common center \(C\), \(A/B\) is \(H\)-separable and \(B\) is a direct summand of \(A\) as a \((B,B)\)-bimodule if and only if \(V_A(B)\) is \(C\)-Azumaya and \(A\cong B\otimes_CV_A(B)\), where \(V_A(B)\) denotes the centralizer of \(B\) in \(A\). This result extends the commutator theorem of \textit{F. DeMeyer} and \textit{E. Ingraham} [Separable algebras over commutative rings, Lect. Notes Math. 181 (1971; Zbl 0215.36602)], which says that if \(A/C\) is a central separable extension, then for any central separable subalgebra \(B\), \(A\cong B\otimes V_A(B)\). They then discuss the relation between Hopf-Galois extensions and \(H\)-separable extensions. They give a necessary and sufficient condition for a Hopf-Galois extension to be \(H\)-separable and obtain some equivalent conditions for an \(H\)-separable extension to be Hopf-Galois.