Let R be a commutative
ring with identity. In this paper we propose an extension of the concept
of central separable R-algebra and use this to define an associated Brauer
group for R which contains the classical Brauer group as a subgroup. The
essential difference between our notion of central separable algebra and the
classical one is that we do not require that the algebra have an identity. As a
consequence, our algebras need not be finitely generated or projective as
R-modules. Nevertheless, with equality defined using an appropriate version of
Morita equivalence and tensor product providing the operation, we obtain a
tractable extension of the Brauer group. If R is a Henselian local ring with
algebraically closed residual field, our Brauer group is trivial. If R is the
algebra of complex valued continuous functions on a compact Hausdorff
space our Brauer group is the full integral third Čech cohomology group of
the underlying space, while the classical Brauer group is just the torsion
subgroup.
