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.