In mathematics, an Azumaya algebra is a generalization of central simple algebras to R-algebras where R need not be a field. Such a notion was introduced in a 1951 paper of Goro Azumaya, for the case where R is a commutative local ring. The notion was developed further in ring theory, and in algebraic geometry, where Alexander Grothendieck made it the basis for his geometric theory of the Brauer group in Bourbaki seminars from 1964–65. There are now several points of access to the basic definitions. Over a ring An Azumaya algebra over a commutative ring is an -algebra that is finitely generated, faithful, and projective as an -module, such that the tensor product (where is the opposite algebra) is isomorphic to via the map sending to the endomorphism of . Examples over a field Over a field , Azumaya algebras are completely classified by the Artin-Wedderburn theorem since they are the same as central simple algebras. These are algebras isomorphic to the matrix ring for some division algebra over . For example, quaternion algebras provide examples of central simple algebras. Examples over local rings Given a local commutative ring , an -algebra is Azumaya if and only if A is free of positive finite rank as an R-module, and the algebra is a central simple algebra over , hence all examples come from central simple algebras over . Cyclic algebras There is a class of Azumaya algebras called cyclic algebras which generate all similarity classes of Azumaya algebras over a field , hence all elements in the Brauer group (defined below). Given a finite cyclic Galois field extension of degree , for every and any generator there is a twisted polynomial ring , also denoted , generated by an element such that and the following commutation property holds: As a vector space over , has basis with multiplication given by Note that give a geometrically integral variety , there is also an associated cyclic algebra for the quotient field extension . Brauer group of a ring Over fields, there is a cohomological classification of Azumaya algebras using Étale cohomology. In fact, this group, called the Brauer group, can be also defined as the similarity classes of Azumaya algebras over a ring , where rings are similar if there is an isomorphism of rings for some natural numbers . Then, this equivalence is in fact an equivalence relation, and if