Algebra
《Algebra》 · 第 437 页 · Chapter IX，Section 3「Semisimple Rings」，Theorem 3.6 及其后定名段
The following conditions on a nonzero module $A$ over a ring $R$ are equivalent. (i) $A$ is the sum of a family of simple submodules. (ii) $A$ is the (internal) direct sum of a family of simple submodules. (iii) For every nonzero element $a$ of $A$, $Ra\ne 0$; and every submodule $B$ of $A$ is a direct summand (that is, $A=B\oplus C$ for some submodule $C$). A module that satisfies the equivalent conditions of Theorem 3.6 is said to be semisimple or completely reducible.