  Depth three and finite depth are notions known for subfactors via diagrams
and Frobenius extensions of rings via centralizers in endomorphism towers. From
the point of view of depth two ring extensions, we provide a clear definition
of depth three for a tower of three rings C < B < A. If A = B_C and B | C is a
Frobenius extension, this captures the notion of depth three for a Frobenius
extension. For example we provide an algebraic proof that if B | C is depth
three, then A | C is depth two. 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 an interesting
algebraic theory from the point of view of Galois correspondence theory for the
ring {}_BA_C and coring (A \o_B A)^C with respect to the centralizers A^B and
A^C involving Morita context bimodules, nondegenerate pairings and right
coideal subrings.
