Finite depth and Jacobson–Bourbaki correspondence

Abstract

We introduce a notion of depth three tower C ⊆ B ⊆ A with depth two ring extension A | B being the case B = C . If A = End B C and B | C is a Frobenius extension with A | B | C depth three, then A | 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 contained in H . For a depth three tower of rings, a pre-Galois theory for the ring End A C B and coring ( A ⊗ 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.