Basic question in the radical of an ideal

Question

Does the radical of an ideal $\sqrt{I}$ always contain $I$? I think the question boils down to if we are given $\sqrt{I}$, does it contain all the elements such that raised to ANY power? So $x \in \sqrt{I}$ implies $x^n \in I$ for all $n$, or given a specific $n$?

Answer 1

The definition is $\sqrt{I} := \{ x\in R \ | \ x^n \in I \text{ for some } n\in \mathbb{N}\}.$ So an element $x\in R$ is in $\sqrt{I}$ if some power of it lies in $I,$ which is trivially true for the elements of $I.$