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. Ojanguren, "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=25494
Separable algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Separable_algebra&oldid=25494
This article was adapted from an original article by L.A. Bokut' (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article