Remark 2

Let K be an algebraically closed field, and let A be an algebra of finite degree over K. By Remark 4 of VIII, p. 122, the algebra A is semisimple if and only if there exist integers $n_1\geqslant 1, . . . , n_r\geqslant 1$ such that A is isomorphic to the algebra $\mathbf{M}_{n_1}(K)\times \cdots \times \mathbf{M}_{n_r}(K)$.