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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,B)-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 extension of 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).
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