Differential Brauer monoids

The differential Brauer monoid of a differential commutative ring R R is defined. Its elements are the isomorphism classes of differential Azumaya R R algebras with operation from tensor product subject to the relation that two such algebras are equivalent if matrix algebras over them, with entry-wise differentiation, are differentially isomorphic. The fine Bauer monoid, which is a group, is the same thing without the differential requirement. The differential Brauer monoid is then determined from the fine Brauer monoids of R R and R D R^D and the submonoid of the Brauer monoid whose underlying Azumaya algebras are matrix rings.