“…A Maschke-type theorem on certain Hopf algebroids can be obtained also by the application of [37,Theorem 4.2]. Notice, however, that the Hopf algebroids occurring this way are only the Frobenius Hopf algebroids (discussed in Section 4 below), that is the Hopf algebroids possessing nondegenerate integrals (which are called Frobenius integrals in [37] The following assertions on a Hopf algebroid A = (A L , A R , S) are equivalent: [17,Proposition 2.6] that a separable extension is both left and right semi-simple.…”