On free quadratic extensions of rings

The quaternion algebraBj] over a commutative ringB with 1 defined byS. Parimala andR. Sridharan is generalized in two directions: (1) the ringB may be non-commutative with 1, and (2)j ² may be any invertible element (not necessarily –1). LetG={} be an automorphism group ofB of order 2, andA={b inB| (b)=b}. LetBj] be a generalized quaternion algebra such thataj (a) for eacha inB. It will be shown thatB is Galois (for non-commutative ring extensions) overA which is contained in the center ofB if and only ifBj] is Azumaya overA. Also,Aj] is a splitting ring forBj] such thatAj] is Galois overA. Moreover, we shall determine which automorphism group ofAj] is a Galois group.