nilpotent ideal
In mathematics, more specifically ring theory, an ideal I of a ring R is said to be a nilpotent ideal if there exists a natural number k such that I{{Hair space}}k = 0.{{sfn|Isaacs|1993|p=194}} By I{{Hair space}}k, it is meant the additive subgroup generated by the set of all products of k elements in I.{{sfn|Isaacs|1993|p=194}} 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.{{sfn|Isaacs|1993|loc=Theorem 14.38, p. 210}}{{sfn|Herstein|1968|loc=Theorem 1.4.5, p. 37}} The notion of a nilpotent ideal, although interesting in the case of commutative rings, is most interesting in the case of noncommutative rings.
Relation to nil ideals
The notion of a nil ideal has a deep connection with that of a nilpotent ideal, and in some classes of rings, the two notions coincide. If an ideal is nilpotent, it is of course nil, but a nil ideal need not be nilpotent for more than one reason. The first is that there need not be a global upper bound on the exponent required to annihilate various elements of the nil ideal, and secondly, each element being nilpotent does not force products of distinct elements to vanish.{{sfn|Isaacs|1993|p=194}}
In a right Artinian ring, any nil ideal is nilpotent.{{sfn|Isaacs|1993|loc=Corollary 14.3, p. 195}} 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.{{sfn|Herstein|1968|loc=Theorem 1.4.5, p. 37}}
See also
Notes
{{Reflist}}
References
- {{cite book
| first = I.N.
|last=Herstein
| year = 1968
| title = Noncommutative rings
| edition =1st
| publisher = The Mathematical Association of America
| isbn = 0-88385-015-X
}}
- {{cite book
| first = I. Martin
|last=Isaacs
| author-link = Martin Isaacs
| year = 1993
| title = Algebra, a graduate course
| edition = 1st
| publisher = Brooks/Cole Publishing Company
| isbn = 0-534-19002-2
}}