A NOTE ON OUTER GALOIS THEORY OF RINGS

Let G be a finite group of automorphisms of a ring B, and let A be the subring of G-invariant elements of B. Call B an outer semi-Galois extension of A, if the centralizer of A in B is the center of B and B is a separable extension of A (i.e., the (B, 2?)-bimodule homomorphism of B ⊗AB onto B, which is determined by the ring multiplication in B, splits). The principal result of this paper is more easily stated here under the additional hypothesis that A is a direct summand of the right A-module B. THEOREM. If B is an outer semi-Galois extension of a subring A0 and A0 is a direct summand of the right A0-module B, then the following statements are equivalent for an intermediate ring A. (1) B is an outer semi-Galois extension of A and A is a direct summand of the right A-module B. (2) B is a projective Frobenius extensionof A. (3) A is the subring of invariant elements of B with respect to a finite group of automorphisms of B (not necessarily a subgroup of G). © 1969 by Pacific Journal of Mathematics.