A Nil Ideal that is not Nilpotent
An ideal is said to be nil is each of elements is nilpotent. is said to be nilpotent if there exists some such that That is, every product of elements of is zero.
Consider the ring . Then, the ideal is nil in because each of its generators is nilpotent.
Moreover, is not nilpotent. To see this, assume by contradiction that it is. Then, there exists some such that This implies that for all in . However, in . This is the desired contradiction.