Definition 1

We say that an algebra A over the field K is central if the mapping $\lambda \mapsto \lambda 1$ is a bijection from K to the center of A.

A central algebra is not reduced to 0. For every integer $n\geqslant 1$, the matrix K-algebra $\mathbf{M}_n(K)$ is central (VIII, p. 83, Corollary 2) and simple (VIII, p. 120, Theorem 1). More generally, let D be a central K-algebra of finite degree; then $\mathbf{M}_n(D)$ is also central. Let A be a simple ring; its center Z is a field (VIII, p. 121, Corollary 1), and A is therefore a central simple algebra over Z. If the field K is algebraically closed, then a simple algebra of finite degree over K is central (VIII, p. 122, Corollary 3). The opposite algebra of a central simple algebra is central simple.