Radical ideal

Remarks

Example: $6\Z$ is a radical ideal of $\Z$; if $6$ divides $r^n$, then $2$ and $3$ divide $r$, so $6$ divides $r$. Non-example: $12\Z$ is not radical, since $6^2 \in 12\Z$ and $6 \notin 12\Z$. Every prime ideal is radical: if $r^n \in P$, primality applied $n - 1$ times gives $r \in P$. The ideal $I$ is radical if and only if the quotient ring $R/I$ is reduced. The radical $\sqrt{J}$ of any ideal $J$ is a radical ideal.