Finite depth and Jacobson–Bourbaki correspondence

We introduce a notion of depth three tower Csubset of or equal toBsubset of or equal toA with depth two ring extension AB being the case B=C. If image and BC is a Frobenius extension with ABC depth three, then AC is depth two. If A, B and C correspond to a tower G>H>K via group algebras over a base ring F, the depth three condition is the condition that K has normal closure KG contained in H. For a depth three tower of rings, a pre-Galois theory for the ring image and coring (Acircle times operatorBA)C involving Morita context bimodules and left coideal subrings is applied to specialize a Jacobson–Bourbaki correspondence theorem for augmented rings to depth two extensions with depth three intermediate division rings.