An Isomorphism Problem for Azumaya Algebras with Involution over Semilocal Bézout Domains

Let R be a semilocal Bezout domain with fraction field F. Assume that 2 is invertible in R. The main result of this article states that R−algebras with involution that become isomorphic over F are already isomorphic over R. We also show that this implies that hermitian or skew-hermitian spaces over an R−algebra with involution without zero divisors that become similar over F, are already similar over R.