On strongly separable Frobenius extensions (Q1355166)

The author gives a new proof of the characterization of strong separabilities as given by Mewborn-McMahon and Yamashiro; furthermore he gives an additional characterization in terms of a centrally projective bimodule. He uses this new approach to prove in the second part of the paper that every Frobenius, strongly separable extension \(A/B\) gives an \(H\)-separable Frobenius extension \(A/B^*\), where \(B^*\) is the double commutator of \(B\) in \(A\). In the process he shows that for a Frobenius extension \(A/B\), the strong separability condition of \(A\) over \(B\) is equivalent to the Frobenius condition of \(D\) (the commutator of \(B\) in \(A\)) over the center of \(A\). This answers in the Frobenius case the main conjecture: If \(A\) is strongly separable over \(B\), then \(A\) is \(H\)-separable over \(B^*\) and \(B^*\) is separable over \(B\).