r+I=0 in R/I if and only if r is an element of I. Think nilpotent in the first case and radical in the second.
Prove that is radical if and only if has no nilpotent elements.
First: Should the problem state "Prove that is radical if and only if has no nonzero nilpotent elements."?
An ideal I is called radical if such that (for some ), then also.
Note: Rings are defined to be associative and commutative with unity.