Depth Two, Separability and a Trace Ideal Condition for Frobenius Extensions

To a depth two extension ⊇ there are associated ring and coring constructs called bialgebroids in a duality-for-actions set-up which coalesce into a Jones tower when the extension is Frobenius [12, Kadison-Szlachányi]. A number of the ring and module constructs are perfectly well-defined for any ring extension; in particular, a generalized Miyashita-Ulbrich action of the -central tensor square acting from the right on the centralizer, introduced in this paper. From this point of view, we first review, then extend the depth two theory in this paper including a discussion of classical examples in group theory. Then we apply some of the same constructs to a study of separability and Frobenius extensions. We provide new characterizations of separable and H-separable extensions that show similarities with depth two extensions. With a view to the problem of when separable extensions are Frobenius, we then provide a trace ideal condition for when a ring extension is Frobenius