In Section 3, we prove a relative Maschke theorem characterizing semisimple extension of finite-dimensional Hopf algebras as a separable extension; as a corollary, these are ordinary (or untwisted) Frobenius extensions.
More strongly, a ring extension A [??] B is said to be a separable extension if for any right A-module M, the multiplication epimorphism [[micro].sub.M] : M [[cross product].sub.B] A [right arrow] M splits , which also generalizes the straightforward notion of left semisimple extension.
The Hopf subalgebra pair R [??] H is a right (or left) semisimple extension [??] [k.sub.H] | [Q.sub.H] [??] [k.sub.H] is R-relative projective [??] there is q [member of] Q such that [[epsilon].sub.Q] (q) [not equal to] 0 and qh = q[epsilon](h) for every h [member of] H [??] [there exists] s [member of] H : s[H.sup.+] [??] [R.sup.+] H and [epsilon](s) = 1 [??] H is a separable extension of R.
if the extension A [??] B is a separable, then R [??] S is a separable extension ;