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I'm trying to show the intersection of radical ideals is radical. Let $A$ and $B$ be radical ideals, and let $x\in\text{ rad } (A\cap B)$. Then there is an $n\in\mathbb{N}$ such that $x^n\in A\cap B$. Where do I go from here?
$x^n\in A\cap B\subset A$ implies that $x\in A$ since $A$ radical, the same argument shows $x\in B$.
• $x^n\in A\cap B\subset A$ implies that $x\in A$ since $A$ radical, the same argument shows $x\in B$.