Finite depth and Jacobson-Bourbaki correspondence

We introduce a notion of depth three tower C C B C A with depth two ring extension A vertical bar B being the case B = C. If A = End B-C and B vertical bar C is a Frobenius extension with A vertical bar B vertical bar C depth three, then A vertical bar C 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 K-G containedin H. For a depth three tower of rings, a pre-Galois theory for the ring End (B)A(C) andcoring (A circle times(B) A)(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. (C) 2007 Elsevier B.V. All rights reserved.