Separable algebra

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2020 Mathematics Subject Classification: Primary: 16-XX [MSN][ZBL]

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 $A \otimes_R A^\textrm{o} = A^e$-module (cf. Projective module). Here, $A^\textrm{o}$ 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.


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How to Cite This Entry:
Separable algebra. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by L.A. Bokut' (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article