central simple algebra
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 $D$.
Title  central simple algebra 

Canonical name  CentralSimpleAlgebra 
Date of creation  20130322 11:49:08 
Last modified on  20130322 11:49:08 
Owner  djao (24) 
Last modified by  djao (24) 
Numerical id  7 
Author  djao (24) 
Entry type  Definition 
Classification  msc 16D60 
Classification  msc 70K75 