central simple algebra

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

  • A has an identity elementMathworldPlanetmath, as a ring

  • A is central: the center of A equals K (for all zA, we have za=az for all aA if and only if zK)

  • A is simple: for any two sided ideal I of A, either I={0} or I=A

By a theoremMathworldPlanetmath of Brauer, for every central simple algebra A over K, there exists a unique (up to isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath) division ring D containing K and a unique natural numberMathworldPlanetmath n such that A is isomorphic to the ring of n×n matrices with coefficients in D.

Title central simple algebra
Canonical name CentralSimpleAlgebra
Date of creation 2013-03-22 11:49:08
Last modified on 2013-03-22 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