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?
Nilpotent elements in a quotient ring.
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abstract-algebra
ideals
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2The 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