Assume \(R\) is commutative. Prove that the set of prime ideals in \(R\) has a minimal element with respect to inclusion (possibly the zero ideal).
The prime ideals form a partially ordered set \(S\) by inclusion. If \(C\) is a chain in \(S\), then define \(P = \cap_{I \in C} I\). Suppose \(ab \in P\) and \(a,b \not\in P\). Then there is some \(I_a, I_b \in C\) such that \(a \not\in I_a\) and \(b \not\in I_b\). If \(I_a \subseteq I_b\), then \(b \not\in I_a\), contradicting the fact that \(I_a\) is a prime ideal (likewise if \(I_b \subseteq I_a\)). Thus \(P\) is a prime ideal and a lower bound for \(C\). By Zorn's Lemma, there exists a minimal prime ideal.
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