Separable algebra

separable algebra over a field

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

separable algebra over a ring

An algebra $A$ over a commutative ring $R$ is separable if $A$ is projective as a left $A \otimes_R A^o = A^e$-module (cf. Projective module). Here, $A^o$ is the opposite algebra of $A$.

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

[Wae]	B.L. van der Waerden, "Algebra" , 1–2 , Springer (1967–1971) (Translated from German)
[CuRe]	C.W. Curtis, I. Reiner, "Representation theory of finite groups and associative algebras" , Interscience (1962)
[AuGo]	M. Auslander, O. Goldman, "The Brauer group of a commutative ring" Trans. Amer. Math. Soc. , 97 (1960) pp. 367–409
[MeIn]	F. de Meyer, E. Ingraham, "Separable algebras over commutative rings" , Lect. notes in math. , 181 , Springer (1971)
[KnuOj]	M.-A. Knus, M. Ojanguren, "Théorie de la descente et algèbres d'Azumaya" , Lect. notes in math. , 389 , Springer (1974)
[CaOy]	S. Caenepeel, F. van Oystaeyen, "Brauer groups and the cohomology of graded rings" , M. Dekker (1988)