3. Existence results

These use Zorn's Lemma (which depends on the Axiom of Choice): In a partially ordered set in which every chain has an upper bound (not necessarily in the chain), there is a maximal element.

Proposition. Every proper ideal $I$ of $R$ can be extended to a maximal ideal.

Proof. Apply Zorn to the partially ordered set ${\cal P}$ consisting of the set of proper ideals that contain $I$. Given a chain, take the union of all its members, which is again an ideal in ${\cal P}$. Here it is important that the ring has an identity 1; without that, the union of a chain of proper ideals might not be proper.

Proposition. For any ideal $I$ of $R$, the intersection of all prime ideals containing $I$ is $\sqrt I$ (the radical of $I$, meaning the ideal $\{x \in R : x^n \in I$    for some $n \}$).

Proof. Exercise using Zorn.