Vol. 64, No. 1, 1976

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Infinite Galois theory for commutative rings

John E. Cruthirds

Vol. 64 (1976), No. 1, 107–117

Let S be a commutative ring with identity. A group G of automorphisms of S is called locally finite, if for each s S, the set {σ(s) : σ G} is finite. Let R be the subring of G-invariant elements of S. An R-algebra T is called locally separable if every finite subset of T is contained in an R-separable subalgebra of T. For an R-separable subalgebra T of S and for G a locally finite group of automorphisms it is shown that T is the fixed ring for a group of automorphisms of S. If, in addition, it is assumed that S has finitely many idempotent elements, then it is shown that any locally separable subring T of S is the fixed ring for a locally finite group of automorphisms of S. Examples are included which show the scope of these theorems.

Mathematical Subject Classification 2000
Primary: 13B05
Received: 16 December 1975
Published: 1 May 1976
John E. Cruthirds