On characterizations of a center Galois extension

Abstract

Let B be a ring with 1, C the center of B, G a finite automorphism group of B, and BG the set of elements in B fixed under each element in G. Then, it is shown that B is a center Galois extension of BG (that is, C is a Galois algebra over CG with Galois group G|C≅G) if and only if the ideal of B generated by {c−g(c)|c∈C} is B for each g≠1 in G. This generalizes the well known characterization of a commutative Galois extension C that C is a Galois extension of CG with Galois group G if and only if the ideal generated by {c−g(c)|c∈C} is C for each g≠1 in G. Some more characterizations of a center Galois extension B are also given.

Keywords

• Galois extensions
• center Galois extensions
• central extensions
• Galois central extensions
• Azumaya algebras
• separable extensions
• H-separable extensions.