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Let $R$ be a commutative ring with 1. Give an example in which the radical of infinitely many ideals is not equal to the intersection of the radicals.

user101689
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1 Answers1

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$R=k[x]$, $k$ a field, and $a_n=(x^n)$, $n\ge 1$.

  • More generally, if $I$ is an ideal of a commutative ring $R$, then $\cap_n \sqrt{I^n} = I$ is usually larger than $\sqrt{\cap_n I^n}$, for example when $\cap_n I^n = 0$. – Martin Brandenburg Oct 18 '13 at 16:01