Let \(A\) be a commutative, noetherian ring. Nagata’s criterion gives an axiomatic procedure for proving that some geometric loci of Spec\((A)\) are open. The author shows directly that Nagata’s criterion applies to the loci of primes satisfying Serre’s \((R_n)\) or \((S_n)\) regularity conditions (for all \(n\geq 0\)), and hence these loci are open in Spec(\(A\)).