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

• $A$ has an identity element, as a ring
• $A$ is central: the center of $A$ equals $K$ (for all $z \in A$ , we have $z\cdot a = a \cdot z$ for all $a \in A$ if and only if $z \in K$ )
• $A$ is simple: for any two sided ideal $I$ of $A$ , either $I = \{0\}$ or $I = A$

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