Hopf module duality applied to X-outer Galois theory

Abstract

We work over a fixed ground field k in arbitrary characteristic. Let R be a prime algebra, and let Q denote the symmetric Martindale quotient ring of R. We know from [10, Section 6.4] that to study group actions on R, it is fundamental to consider the extended actions on Q, and those actions which are X-outer, or outer as automorphisms of Q, have nice properties, which could be generalized for X-outer (in some appropriate sense) Hopf algebra actions. To recall Kharchenko’s Galois-type correspondence theorem, let G be a finite group of X-outer automorphisms of R. It was proved in [2, Theorem 3.10.2], [11, Theorem B] that