Mathematics Asked on December 22, 2020

So suppose that we have already shown that the radical of an ideal $I$, $sqrt{I}$, is an ideal. Can we just conclude that the $sqrt{I}$ is a radical ideal, as it is i) radical of an ideal, ii) ideal?

It depends on your definition of "radical ideal". If it is:

An ideal $J$ is radical if and only if there is an ideal $I$ such that $J=sqrt I$.

Then, yes. If it is the more common version:

An ideal $J$ is radical if and only if $sqrt J=J$.

Then, no: you still have to prove that $sqrt{sqrt{I}}=sqrt I$. The two definitions are equivalent, pretty much because of the identity $sqrt{sqrt{I}}=sqrt I$.

Correct answer by Gae. S. on December 22, 2020

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