# Separable algebra

From Encyclopedia of Mathematics

A finite-dimensional semi-simple associative algebra over a field that remains semi-simple under any extension of (that is, the algebra is semi-simple for any field , cf. Semi-simple algebra). An algebra is separable if and only if the centres of the simple components of this algebra (see Associative rings and algebras) are separable extensions of (cf. Separable extension).

#### References

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

[2] | C.W. Curtis, I. Reiner, "Representation theory of finite groups and associative algebras" , Interscience (1962) |

#### Comments

An algebra over a commutative ring is separable if is projective as a left -module (cf. Projective module).

An algebra that is separable over its centre is called an Azumaya algebra. These algebras are important in the theory of the Brauer group of a commutative ring or scheme.

#### References

[a1] | M. Auslander, O. Goldman, "The Brauer group of a commutative ring" Trans. Amer. Math. Soc. , 97 (1960) pp. 367–409 |

[a2] | F. de Meyer, E. Ingraham, "Separable algebras over commutative rings" , Lect. notes in math. , 181 , Springer (1971) |

[a3] | M.-A. Knus, M. Ojangouren, "Théorie de la descente et algèbres d'Azumaya" , Lect. notes in math. , 389 , Springer (1974) |

[a4] | S. Caenepeel, F. van Oystaeyen, "Brauer groups and the cohomology of graded rings" , M. Dekker (1988) |

**How to Cite This Entry:**

Separable algebra.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Separable_algebra&oldid=13896

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