Proposition 4

Let A be a ring that has a minimal left ideal $\mathfrak{a}$. Every faithful simple A-module is isomorphic to the A-module $\mathfrak{a}$.

Let M be a faithful simple A-module. Let $\alpha$ be a nonzero element of $\mathfrak{a}$. Since the A-module M is faithful, there exists an element $x$ of M such that $\alpha x\not= 0$. The homomorphism $a\mapsto ax$ from $\mathfrak{a}$ to M is then nonzero; it is therefore an isomorphism by Proposition 2 of VIII, p. 47.