Algebra
《Algebra》 · 第 435—436 页 · Chapter IX，Section 3，Theorem 3.3「Wedderburn-Artin」
Theorem 3.3. (Wedderburn-Artin). The following conditions on a ring $R$ are equivalent. (i) $R$ is a nonzero semisimple left Artinian ring; (ii) $R$ is a direct product of a finite number of simple ideals each of which is isomorphic to the endomorphism ring of a finite dimensional vector space over a division ring; (iii) there exist division rings $D_1,\ldots,D_t$ and positive integers $n_1,\ldots,n_t$ such that $R$ is isomorphic to the ring $\operatorname{Mat}_{n_1}D_1\times\operatorname{Mat}_{n_2}D_2\times\cdots\times\operatorname{Mat}_{n_t}D_t$.