Intersection of radical ideal

2017-12-03 03:07:56

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 $x\in B$.

    2017-12-03 04:37:15