When is the ring of 2x2 matrices over a ring Galois?

Abstract

Let R be a ring with 1, M2(R) the ring of 2 × 2 matrices over R, and S the Sugano quaternion ring over R. Then, S is isomorphic with a subring of M2(R) if and only if 2 is not a zero divisor in R, and S ∼ M2(R) if and only if 2 is invertible in R. Moreover, if 2 is invertible in R, then M2(R) is a Galois extension of R with an abelian inner Galois group G of order 4. This implies that the rings of 2 × 2 matrices over the real field and complex field are central Galois algebras induced by the central Galois algebra of 2 × 2 matrices over the rational field with Galois group G. Mathematics Subject Classification: 16S35, 16W20