---
tag: 99T7Z4T39W
kind: definition
name: Central simple algebra
msc: [16K20, 16K50]
defines: 5HRKJBAY2V
uses: [04C5ZBG1WT, 3CTW1P4HBP, 62M0NC3E2T, MVT64B3B6H]
sources:
  - url: https://en.wikipedia.org/wiki/Central_simple_algebra
---

## Definition

Let $K$ be a [[field|04C5ZBG1WT]].
A *central simple algebra* over $K$ is an [[associative algebra|3CTW1P4HBP]] $A$ over $K$ with these three properties:

1. $A$ is finite-dimensional as a vector space over $K$.
2. $A$ is a [[simple ring|62M0NC3E2T]]: $A \ne 0$ and its only two-sided ideals are $0$ and $A$.
3. The [[center|MVT64B3B6H]] of $A$ is $K \cdot 1$.

## Remarks

Example: $M_n(K)$ is a central simple algebra over $K$ for $n \ge 1$.
It has dimension $n^2$, its only two-sided ideals are $0$ and itself, and a matrix that commutes with all matrix units $E_{ij}$ is a scalar matrix.

Non-example: $\C$ as an algebra over $\R$ is finite-dimensional and simple, but its center is $\C \ne \R$, so clause 3 fails.
Non-example: $K \times K$ is commutative of dimension $2$ and has the ideal $K \times 0$, so clause 2 fails.

Hamilton's quaternions $\mathbb{H}$ are a central simple algebra over $\R$ of dimension $4$.
The dimension of a central simple algebra is always a square $n^2$, and $n$ is its degree.
If $A$ and $B$ are central simple over $K$, so is $A \otimes_K B$; and $A \otimes_K A^{\mathrm{op}} \cong M_{n^2}(K)$ when $\dim_K A = n^2$.