In mathematics, more specifically ring theory, an ideal, *I*, of a ring is said to be a **nilpotent ideal**, if there exists a natural number *k* such that *I**k* = 0. By *I**k*, it is meant the additive subgroup generated by the set of all products of *k* elements in *I*. Therefore, *I* is nilpotent if and only if there is a natural number *k* such that the product of any *k* elements of *I* is 0.

The notion of a nilpotent ideal is much stronger than that of a nil ideal in many classes of rings. There are, however, instances when the two notions coincide—this is exemplified by Levitzky's theorem. The notion of a nilpotent ideal, although interesting in the case of commutative rings, is most interesting in the case of noncommutative rings.

Read more about Nilpotent Ideal: Relation To Nil Ideals, See Also

### Famous quotes containing the word ideal:

“The archetype of all humans, their *ideal* image, is the computer, once it has liberated itself from its creator, man. The computer is the essence of the human being. In the computer, man reaches his completion.”

—Friedrich Dürrenmatt (1921–1990)