Abstract
The quaternion algebraB[j] 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={ϱ} be an automorphism group ofB of order 2, andA={b inB|ϱ (b)=b}. LetB[j] be a generalized quaternion algebra such thataj ϱ (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 ifB[j] is Azumaya overA. Also,A[j] is a splitting ring forB[j] such thatA[j] is Galois overA. Moreover, we shall determine which automorphism group ofA[j] is a Galois group.
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Szeto, G., Wong, YF. On free quadratic extensions of rings. Monatshefte für Mathematik 92, 323–328 (1981). https://doi.org/10.1007/BF01320063
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DOI: https://doi.org/10.1007/BF01320063