# Central algebra

From Encyclopedia of Mathematics

An algebra with a unit element over a field, the centre of which (see Centre of a ring) coincides with the ground field. For example, the division ring of quaternions is a central algebra over the field of real numbers, but the field of complex numbers is not. The full matrix algebra over a field is a central algebra. The tensor product of a simple algebra and a central simple algebra is a simple algebra, which is central if and only if the first one is. Every automorphism of a finite-dimensional central simple algebra is inner and its dimension is the square of an integer.

#### References

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

[2] | L.A. Skornyakov, "Elements of general algebra" , Moscow (1983) (In Russian) |

[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 algebra.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Central_algebra&oldid=53620

This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article