Existence of maximal ideals

Existence of maximal ideals (Zorn): Let $R$ be a unital ring with $1\neq 0$, assumed commutative. Then $R$ has a maximal ideal .

This result is typically proved using Zorn's lemma (and hence the axiom of choice ) applied to the partially ordered set of proper ideals of $R$ ordered by inclusion.

Proof sketch: Let $\mathcal{P}$ be the set of proper ideals of $R$, ordered by inclusion. Any chain $\mathcal{C}\subseteq\mathcal{P}$ has an upper bound $\bigcup_{J\in\mathcal{C}}J$, which is an ideal and remains proper because $1\notin J$ for all $J\in\mathcal{C}$. By Zorn, $\mathcal{P}$ has a maximal element, which is a maximal ideal.