Group actions on von Neumann regular rings (Q1333952)

Let \(A\) be a ring with identity and \(G\) be a finite group of automorphisms of \(A\). Then it is well known that \(A\) can be a left (or right) module over the skew group ring \(A*G\). The author shows that \(A\) is a Galois extension of the fixed ring \(A^ G\) if and only if \(A\) is a Frobenius extension of \(A^ G\) and the module \({}_{A*G} A\) (or \(A_{A*G}\)) is faithful. Moreover it is shown that if \(A\) is von Neumann regular and \(| G|\) is invertible in \(A\), then the following conditions are equivalent: (1) \(A\) is biregular self-injective; (2) \(A^ G\) is biregular self-injective and \(A\) is a Frobenius extension of \(A^ G\); (3) \(A*G\) is biregular self-injective. Thereby this result improves Theorem 8 in [\textit{D. Handelman} and \textit{G. Renault}, Pac. J. Math. 89, 69-80 (1980; Zbl 0449.16009)] and Theorem 8 in [\textit{S. Jøndrup}, J. Lond. Math. Soc., II. Ser. 8, 483-486 (1974; Zbl 0284.13007)].