Nilpotent ideal

[1] By I k, it is meant the additive subgroup generated by the set of all products of k elements in I.

There are, however, instances when the two notions coincide—this is exemplified by Levitzky's theorem.

[1] In a right Artinian ring, any nil ideal is nilpotent.

[4] This is proven by observing that any nil ideal is contained in the Jacobson radical of the ring, and since the Jacobson radical is a nilpotent ideal (due to the Artinian hypothesis), the result follows.

In fact, this can be generalized to right Noetherian rings; this result is known as Levitzky's theorem.