Theorem 1 (Krull)

Let $A$ be a ring and $a$ a left ideal of $A$ distinct from $A$. There exists a maximal ideal $m$ of $A$ containing $a$.

Consider $A$ as operating on the additive group $A^+$ of $A$ by left multiplication. Then the left ideals of $A$ are the stable subgroups of $A^+$. The theorem thus follows from § 4, no. 3, Proposition 3 applied to the subset $P = \{1\}$ of $A^+$.