= Finite prime-intersection representation of a radical
{title2=$\sqrt I=P_1\cap\cdots\cap P_s$}
In a <Noetherian ring>, every <radical of an ideal> is an intersection of finitely many <prime ideals>. A maximal counterexample <ideal> $J$ cannot be prime. Choose $a,b\notin J$ with $ab\in J$; the strictly larger <ideals> $J+(a)$ and $J+(b)$ satisfy the result, and $\sqrt J=\sqrt{J+(a)}\cap\sqrt{J+(b)}$. This contradicts maximality. The unit <ideal> uses an empty intersection.
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