What is the difference between radical ideal and the radical of an ideal?

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?

One Answer

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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