# Central simple algebra

A simple associative algebra with a unit element that is a central algebra. Every finite-dimensional central simple algebra over a field is isomorphic to a matrix algebra over a finite-dimensional central division algebra over . In particular, if is algebraically closed, then every finite-dimensional central simple algebra over is isomorphic to , and if , then is isomorphic to the algebra of real or quaternion matrices. The tensor product of a central simple algebra and an arbitrary simple algebra is a simple algebra, which is central if is central. Two finite-dimensional central simple algebras and over are called equivalent if

for certain and , or, which is equivalent, if and are isomorphic matrix algebras over one and the same central division algebra. The equivalence classes of central simple algebras over form the Brauer group of relative to the operation induced by tensor multiplication.

#### References

[1] | B.L. van der Waerden, "Algebra" , 1–2 , Springer (1967–1971) (Translated from German) |

[2] | Yu.A. Drozd, V.V. Kirichenko, "Finite-dimensional algebras" , Kiev (1980) (In Russian) |

#### Comments

#### References

[a1] | R.S. Peirce, "Associative algebras" , Springer (1980) |

[a2] | A.A. Albert, "Structure of algebras" , Amer. Math. Soc. (1939) |

[a3] | M. Deuring, "Algebren" , Springer (1935) |

[a4] | I.N. Herstein, "Noncommutative rings" , Math. Assoc. Amer. (1968) |

[a5] | N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956) |

**How to Cite This Entry:**

Central simple algebra.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Central_simple_algebra&oldid=13839