Study of the quotient module of a finite-dimensional Hopf subalgebra pair in order to compute its depth yields a relative Maschke Theorem, in which semisimple extension is characterized as being separable, and is therefore an ordinary Frobenius extension.

H is shown in [31] to be precisely determined by the depth of their quotient module [Q.

For example, a finite group algebra extension has quotient module Q equal to a permutation module, which is algebraic [13, Ch.

We note that the length of the annihilator chain of tensor powers of the quotient module is linearly related to the depth if the Hopf algebra is semisimple, improving on some results in [15].

We study the quotient module Q of a finite-dimensional Hopf subal-gebra pair R [?

cross product](n)] of the quotient module of the Hopf subalgebra pair [R.

We continue the study begun in [15] relating the depth of a quotient module Q to its descending chain of annihilator ideals of its tensor powers:

H/I is a Hopf subalgebra pair with quotient module isomorphic to Q by a Noether isomorphism theorem.

It is interesting at this point to compute the quotient module Q for the inclusion C [S.