Radical and radical ideal

Question

I am trying to get used to the terminology of commutative algebra. Let $R$ be a commutative ring with unity. Then I know radical $\sqrt{I} := \{ a\in R| a^n \in I, \textrm{for some $n \in \mathbb{Z}$} \}$ and an ideal $I$ is called a radical ideal if $I=\sqrt{I}$.

Since $\sqrt{I}$ itself is a ideal containing $I$, I am confused with the terminology a "radical ideal". Radical and Radical ideal are different. Am I right?

It seems this radical ideals are important tools for Hilbert's many theorem thus important to commutative algebras and algebraic geometry. So there seems to reason the mathematician defined "radical ideals".

I know some property of radical and radical ideals for prime ideal. For example, for prime ideal $P$, $\sqrt{P} =P$. so Every prime ideal is a radical ideal.

Answer 1

I'm not entirely sure what confuses you, but the point is that you can define "radical ideals" in two equivalent ways:

• an ideal $J$ is a radical ideal if $\sqrt{J}=J$
• an ideal $J$ is a radical ideal if $J = \sqrt{I}$ for some ideal $I$ (in other words $J$ is the radical of some ideal).

This is the same thing because $\sqrt{\sqrt{I}}=\sqrt{I}$. So the first definition obviously implies the second one, and the converse is true because if $J=\sqrt{I}$ then $\sqrt{J} = \sqrt{\sqrt{I}} = \sqrt{I}=J$.