On the semisimplicity of twisted group rings and generalized Maschke's theorem

Classical Maschke's theorem gives a condition for the semisimplicity of group algebras over a field. As group algebras are a particular case of twisted group rings, we extend Maschke's theorem to twisted group rings over division rings by proving that if the characteristic of a division ring K does not divide the order of a finite group G, twisted group ring K(G; H) is semisimple. Then we prove the converse of this theorem for twisted group rings arising from split extensions of K by G. As there is a one-to-one correspondence between twisted group rings and group lattices over division rings, we conclude with the consequence of the semisimplicity of twisted group rings on group lattices.