Separable algebra

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A separable algebra is a kind of semisimple algebra. It is a generalization to associative algebras of the notion of a separable field extension.

Contents 1 Definition 2 Commutative separable algebras 3 Examples 4 References

[edit] Definition

Let K be a field. An associative K-algebra A is said to be separable if for every field extension the algebra is semisimple.

[edit] Commutative separable algebras

If is a field extension, then L is separable as an associative K-algebra if and only if the extension of field is separable.

[edit] Examples

If K is a field and G is a finite group such that the order of G is invertible in K, then the group ring K[G] is a separable K-algebra.

[edit] References

• Charles Weibel, Homological algebra