CentralSimpleAlgebra 2001-10-19 01:06:11 2002-02-13 16:02:26 Definition \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{graphicx} \usepackage{xypic} Let $K$ be a field. A {\em central simple algebra} $A$ (over $K$) is an algebra $A$ over $K$, which is finite dimensional as a vector space over $K$, such that \begin{itemize} \item $A$ has an identity element, as a ring \item $A$ 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 $z \in K$) \item $A$ is simple: for any two sided ideal $I$ of $A$, either $I = \{0\}$ or $I = A$ \end{itemize} 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 $D$.