Let be a field. A central simple algebra (over ) is an algebra over , which is finite dimensional as a vector space over , such that

• has an identity element, as a ring
• is central: the center of equals (for all , we have for all if and only if )
• is simple: for any two sided ideal of , either or

By a theorem of Brauer, for every central simple algebra over , there exists a unique (up to isomorphism) division ring containing and a unique natural number such that is isomorphic to the ring of matrices with coefficients in .