Existence of maximal ideals

Statement

Let $R$ be a ring and let $I$ be an ideal of $R$ with $I \neq R$. Then there is a maximal ideal $M$ of $R$ with $I \subseteq M$. In particular, every nonzero ring has a maximal ideal.

Remarks

The proof uses Zorn's lemma. The union of a chain of proper ideals that contain $I$ is again a proper ideal, because it does not contain $1$. The statement for every nonzero ring is equivalent to the axiom of choice, over the other axioms of ZF set theory. The statement holds for left, right and two-sided ideals.

History

Wolfgang Krull proved the theorem in 1929 with transfinite induction.

Explanations

L1 human

Start with a proper ideal and make it larger, step by step, without reaching the whole ring. In an infinite ring this process may not stop by itself. Zorn's lemma guarantees that an ideal that you cannot enlarge exists.