---
tag: X8H7A98Q55
kind: concept
name: Separable algebra
msc: [16H05]
specializes: [3CTW1P4HBP]
sources:
  - url: https://en.wikipedia.org/wiki/Separable_algebra
same_as:
  wikidata: Q2836018
  wikipedia: Separable algebra
---

## Summary

A separable algebra over a commutative ring $K$ is an [[associative algebra|3CTW1P4HBP]] $A$ whose multiplication map $A \otimes_K A^{\mathrm{op}} \to A$ splits as a map of bimodules.
Over a field, for finite-dimensional $A$, it means that $A$ stays [[semisimple|NY25HXA478]] after every extension of the base field.
It generalizes the notion of a [[separable field extension|0WCX7NB4N7]] to noncommutative algebras, and it is the starting point of the theory of Azumaya algebras.

## Explanations

### L1 human

A semisimple algebra can lose its semisimplicity when the base field is enlarged, in the same way as an irreducible polynomial can acquire a repeated root.
Separable algebras are the ones for which this never happens.
For example, $\C$ is separable over $\R$: after tensoring with $\C$ it becomes $\C \times \C$, which is still semisimple.
Over fields of characteristic $0$, and over finite fields, every finite-dimensional semisimple algebra is separable.