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Let R be a ring. Let I be an ideal of R. If R/I doesn't have nonzero nilpotent element, every nilpotent element in R is contained in I. Then, if I contains every nilpotent element in R, there is no nilpotent element in R/I?

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    The desired conclusion holds if and only if $I$ is a *radical ideal*: an ideal $I$ is radical if and only if whenever $a^n\in I$, it follows that $a\in I$.2012-04-19

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