Vol. 31, No. 2, 1969

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A note on the outer Galois theory of rings

Herbert Frederick Kreimer, Jr.

Vol. 31 (1969), No. 2, 417–432
Abstract

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).

Mathematical Subject Classification
Primary: 16.70
Milestones
Received: 16 September 1968
Published: 1 November 1969
Authors
Herbert Frederick Kreimer, Jr.