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An ideal $I$ is said to be radical if $I=\sqrt{I}$. I have 2 questions.

Is it correct ?

$I$ is a radical ideal. $\iff$ If $x^n\in I$ for some $n$, then $x\in I$

$\text{}$ Is it correct ?

Let $J$ be a radical ideal containing $I$. Then $\sqrt{I}\subseteq J$. In other words, $\sqrt{I}$ is the smallest radical ideal containing $I$.

Proof:

Let $x\in\sqrt{I}\implies x^n\in I\subseteq J$. Hence $x\in\sqrt{J}=J$. Thus $\sqrt{I}\subseteq J.$

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    Yes: these two things are correct. – 2017-01-10
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    @random_guy these are correct. but u can add a short proof to the 1st question too. – 2017-01-10

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