Lemma 1

Let $A$ be a ring such that every left ideal of $A$ is finitely generated. Then every nonempty set $\Phi$ of left ideals of $\mathbf{A}$, ordered by inclusion, has a maximal element.

By Zorn's Lemma (Set Theory, III, p. 154, Th. 2) it is enough to prove that $\Phi$ is inductive. Now if $(a_\lambda)$ is a totally ordered family of elements of $\Phi$ then the union $a$ of the ideals $a_i$ is a left ideal of $\mathbf{A}$, and so admits a finite system of generators $(a_i)_{1 \leq i \leq n}$. Since each $a_i$ belongs to an ideal $a_{\lambda_i}$, and since the family $(a_\lambda)$ is totally ordered, the $a_i$ all belong to the largest of the ideals $a_{\lambda_i}$, say $a_1$. Then $a = a_1$ belongs to $\Phi$, which is thus indeed an inductive set.

Later we will study those rings $B$, called noetherian rings, such that every nonempty set of ideals of $B$ contains a maximal element.