Prove that $\displaystyle I$ is radical if and only if $\displaystyle R/I$ has no nilpotent elements.

First: Should the problem state "Prove that $\displaystyle I$ is radical if and only if $\displaystyle R/I$ has nononzeronilpotent elements."?

An ideal I is called radical if $\displaystyle \forall \ x \in R$ such that $\displaystyle x^n \in I$ (for some $\displaystyle n>0$), then $\displaystyle x \in I$ also.

Note: Rings are defined to be associative and commutative with unity.