16H05 18B40 2 groupoid action 2 13B02 1 13B05 1 16G10 1 16G30 1 16S35 1 16W30 1 16W50 1 20L99 1 Azumaya algebras 1 Azumaya ring 1 Galois algebra 1 Galois correspondence 1 Galois-Grothendieck equivalence 1 Gorenstein orders 1 Hopf Galois extensions 1 \(G\)-set 1 and Hirata separable extensions 1 commutator subrings 1 skew groupoid ring 1 smash products 1 strongly graded rings 1 symmetric algebras 1