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Let $R$ be a commutative ring with identity. If $I$ is an ideal of $R$, it's well-known that $I$ is a radical ideal if $I=\sqrt{I}$, where $\sqrt{I}=\{r\in R: r^n\in I\ \mathrm{for\ some\\} n\in\mathbb{Z}^{+}\}$.

I know that every prime ideal is a radical ideal, and also an ideal $I$ is radical if only if $R/I$ is a reduced ring (such ring doesn't have nontrivial nilpotents). So my question is: what other properties satisfy the radical ideals?

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Perhaps the next two most important things are these facts:

  1. If $A$ and $Q$ are ideals of a ring and $Q$ is semiprime, $A^n\subseteq Q$ for some natural number $n$ implies $A\subseteq Q$.

  2. The semiprime ideals are exactly the intersections of sets of prime ideals.

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