View source for Separable algebra

A finite-dimensional semi-simple associative algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084450/s0844501.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084450/s0844502.png" /> that remains semi-simple under any extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084450/s0844503.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084450/s0844504.png" /> (that is, the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084450/s0844505.png" /> is semi-simple for any field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084450/s0844506.png" />, cf. [[Semi-simple algebra|Semi-simple algebra]]). An algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084450/s0844507.png" /> is separable if and only if the centres of the simple components of this algebra (see [[Associative rings and algebras|Associative rings and algebras]]) are separable extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084450/s0844508.png" /> (cf. [[Separable extension|Separable extension]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.L. van der Waerden,   "Algebra" , '''1–2''' , Springer  (1967–1971)  (Translated from German)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C.W. Curtis,   I. Reiner,   "Representation theory of finite groups and associative algebras" , Interscience  (1962)</TD></TR></table>



====Comments====
An algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084450/s0844509.png" /> over a commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084450/s08445010.png" /> is separable if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084450/s08445011.png" /> is projective as a left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084450/s08445012.png" />-module (cf. [[Projective module|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|Brauer group]] of a commutative ring or scheme.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Auslander,   O. Goldman,   "The Brauer group of a commutative ring"  ''Trans. Amer. Math. Soc.'' , '''97'''  (1960)  pp. 367–409</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  F. de Meyer,   E. Ingraham,   "Separable algebras over commutative rings" , ''Lect. notes in math.'' , '''181''' , Springer  (1971)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M.-A. Knus,   M. Ojangouren,   "Théorie de la descente et algèbres d'Azumaya" , ''Lect. notes in math.'' , '''389''' , Springer  (1974)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  S. Caenepeel,   F. van Oystaeyen,   "Brauer groups and the cohomology of graded rings" , M. Dekker  (1988)</TD></TR></table>