Weakly Azumaya algebras

We define a class of algebras, weakly Azumaya algebras, which includes both Azumaya algebras and weak crossed products (cf. Haile [1982, J. Algebra 74, 270-279; 1983, J. Algebra 91, 521-539] and Haile et al. [1983, Amer. J. Math. 105, No. 3, 689-814]). Just as with Azumaya algebras, these have a rank function whose values at localizations of the center are always a square. After a general description of these algebras, we specialize to the case where the center is a field F. The Jacobson radical need not be 0, and we prove a Wedderburn principal theorem for these algebras. Our class is closed under extension of scalars and under tensor products and yields an interesting monoid which generalizes the Brauer group. Our monoid is a union of groups, called stalks, in each of which the unit element is represented by an algebra called an idempotent algebra. The ideal structure of members of the same stalk is the same. A given stalk is not torsion, but the kernel of the restriction map to the algebraic closure is torsion modulo the idempotent algebras. At the end we consider low-dimensional examples in detail.