Central simple algebra

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 $\dim_K A = n^2$.