Example 3

Let K be a field, V a nonzero right vector space over the field K, and A a subring of the ring End$_K(V)$ containing the endomorphisms of V of finite rank (for example, A = End$_K(V)$). Let us prove that V is a simple A-module: let W be a nonzero A-submodule of V and $x$ a nonzero element of W; there exists a linear form $\varphi$ on V such that $\varphi (x)\not= 0$ (II, §7, No. 5, p. 300, Theorem 6). For every $y$ in V, the mapping $z\mapsto y\varphi (z)$, which is linear of rank $\leqslant 1$, belongs to A; we therefore have $Ax= V$, hence a fortiori W = V, which proves that V is a simple A-module.