It's essentially Maschke's theorem

The Galois theory of noncommutative rings is a new subject with roots in invariant theory and in the Galois theory of fields, commutative rings and division rings. If G is a finite group of automorphisms of a ring R, then we are concerned with the relationship between R and the fixed subring R = {re R\r* = r for all g e G}. For the best results it is frequently necessary to assume that R has no |G|-torsion, so that r-|G| = 0 implies r = 0, or even the stronger hypothesis that ICJI e R. A delightful introduction to this material can be found in the survey paper [7] of Fisher and Osterburg. An in-depth study appears in the recent monograph [25] of Montgomery. A useful tool in this subject is the skew group ring RG, the set of all formal sums Z^<=G rgg with rg e R. Addition in RG is componentwise and multiplication is defined distributively by the formula

[1]  M. Lorenz,et al.  Prime ideals in fixed rings II , 1982 .

[2]  D. Passman.  Prime ideals in normalizing extensions , 1981 .

[3]  M. Lorenz.  Finite normalizing extensions of rings , 1981 .

[4]  J. Robson,et al.  Normalizing extensions: Prime ideals and incomparability, II☆ , 1981 .

[5]  D. Passman.  Fixed rings and integrality , 1981 .

[6]  D. Passman,et al.  Prime ideals in crossed products of finite groups , 1979 .

[7]  T. Stafford,et al.  Simple Noetherian rings , 1979 .

[8]  A. Joseph,et al.  An additivity principle for Goldie rank , 1978 .

[9]  S. Montgomery,et al.  Semiprime Skew Group Rings , 1978 .

[10]  S. Montgomery.  The jacobson radical and fixed rings of automorphisms , 1976 .

[11]  S. Montgomery,et al.  Semi-Simple Artinian Rings of Fixed Points , 1975, Canadian Mathematical Bulletin.

[12]  Miriam Cohen.  Semiprime Goldie centralizers , 1975 .

[13]  V. Kharchenko.  Galois extensions and quotient rings , 1974 .

[14]  Thomas W. Hungerford,et al.  The Structure of Rings , 1974 .

[15]  I. Isaacs,et al.  Rings with Fixed-Point-Free Group Actions , 1973 .

[16]  A. Goldie.  The structure of noetherian rings , 1972 .

[17]  O. Villamayor.  On the semisimplicity of group algebras , 1958 .

[18]  D. G. Higman.  Modules with a group of operators , 1954 .

[19]  J. Levitzki.  On Automorphisms of Certain Rings , 1935 .

[20]  H. Maschke.  Ueber den arithmetischen Charakter der Coefficienten der Substitutionen endlicher linearer Substitutionsgruppen , 1898 .