  A minimum depth d^I(S --> R) is assigned to a ring homomorphism S --> R and a
R-R-bimodule I. The recent notion of depth of a subring d(S,R)in a paper by
Boltje-Danz-Kuelshammer is recovered when I = R and S --> R is the inclusion
mapping. Ideal depth gives lower bounds for d(S,R) in case of group C-algebra
pair or semisimple complex algebra extensions. If R | S is a QF extension of
finite depth, minimum left and right even depth are shown to coincide. If R < S
is moreover a Frobenius extension with R a right S-generator, its subring depth
is shown to coincide with its tower depth. In the process formulas for the
ring, module, Frobenius and Temperley-Lieb structures are provided for the
tensor product tower above a Frobenius extension. A depth 3 QF extension is
embedded in a depth 2 QF extension; in turn certain depth n extensions embed in
depth 3 extensions if they are Frobenius extensions or other special ring
extensions with ring structures on their relative Hochschild bar resolution
groups.
