  We introduce a notion of depth three tower of three rings C < B < A as a
useful generalization of depth two ring extension. If A = End B_C and B | C is
a Frobenius extension, this also captures the notion of depth three for a
Frobenius extension in math.RA/0107064 and math.RA/0108067 such that if B | C
is depth three, then A | C is depth two (cf. math.QA/0001020). 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^G contained in H. For a depth three tower of rings, there is a
pre-Galois theory for the ring End {}_BA_C and coring (A \otimes_B A)^C
involving Morita context bimodules and left coideal subrings. This is applied
in the last two sections to a specialization of a Jacobson-Bourbaki
correspondence theorem for augmented rings to depth two extensions with depth
three intermediate division rings.
