Every nonzero ring with identity has a maximal ideal

Let \(R\ne0\) be a ring with identity, and let \(\mathcal P\) be the set of proper ideals of \(R\), ordered by inclusion. Since \((0)\) is proper, \(\mathcal P\) is nonempty.

Let \(\mathcal C\subseteq\mathcal P\) be a chain and set \[ J=\bigcup_{I\in\mathcal C} I. \] Because \(\mathcal C\) is totally ordered by inclusion, \(J\) is an ideal: any two of its elements lie together in one member of the chain, and absorption by elements of \(R\) is inherited from that member. Also \(J\) is proper, since \(1\in J\) would imply \(1\in I\) for some \(I\in\mathcal C\), contradicting \(I\ne R\).

Thus every chain in \(\mathcal P\) has an upper bound in \(\mathcal P\). By Zorn’s lemma, \(\mathcal P\) has a maximal element, which is precisely a proper maximal ideal of \(R\).