On Galois Algebras Satisfying the Fundamental Theorem

Abstract

Let B be a Galois algebra over a commutative ring R with Galois group G such that B ^H is a separable subalgebra of B for each subgroup H of G. Then it is shown that B satisfies the fundamental theorem if and only if B is one of the following three types: (1) B is an indecomposable commutative Galois algebra, (2) B = Re ⊕ R(1 − e) where e and 1 − e are minimal central idempotents in B, and (3) B is an indecomposable Galois algebra such that for each separable subalgebra A, V _B (A) = ⨁∑ _g∈G(A) J _g , and the centers of A and B ^G(A) are the same where V _B (A) is the commutator subring of A in B, J _g  = {b ∈ B | bx = g(x)b for each x ∈ B} for a g ∈ G, and G(A) = {g ∈ G | g(a) = a for all a ∈ A}.

Key Words:

• Azumaya algebras
• Central Galois algebras
• Galois algebras
• Galois extensions
• Separable extensions

2000 Mathematics Subject Classification:

• Primary 13B05, 16W20

ACKNOWLEDGMENT

This article was written under the support of a Caterpillar Fellowship at Bradley University. The authors would like to thank Caterpillar, Inc. for the support.

Notes

Communicated by A. K. Singh.