Finite Depth And

We introduce a notion of depth three tower of three rings C ⊆ B ⊆ A with depth two ring extension A |B recovered when B = C. If A = EndBC and B |C is a Frobenius extension, this captures the notion of depth three for a Frobenius extension in [12, 13] such that if B |C is depth three, then A |C is depth two (a phenomenon of finite depth subfactors, see [20]). We provide a similar definition of finite depth Frobenius extension with embedding theorem utilizing a depth three subtower of the Jones tower. If A, B and C correspond to a tower of subgroups G > H > K via the group algebra over a fixed base ring, the depth three condition is the condition that subgroup K has normal closure K contained in H . For a depth three tower of rings, there is a pre-Galois theory for the ring EndBAC and coring (A⊗BA) C involving Morita context bimodules and left coideal subrings. This is applied in two sections to a specialization of a Jacobson-Bourbaki correspondence theorem for augmented rings to depth two extensions with depth three intermediate division rings.