Summary
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 k = 0.[1] By I k, it is meant the additive subgroup generated by the set of all products of k elements in I.[1] 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.[2][3] 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 edit
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.[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.[3]
See also edit
Notes edit
- ^ a b c Isaacs 1993, p. 194.
- ^ Isaacs 1993, Theorem 14.38, p. 210.
- ^ a b Herstein 1968, Theorem 1.4.5, p. 37.
- ^ Isaacs 1993, Corollary 14.3, p. 195.
References edit
- Herstein, I.N. (1968). Noncommutative rings (1st ed.). The Mathematical Association of America. ISBN 0-88385-015-X.
- Isaacs, I. Martin (1993). Algebra, a graduate course (1st ed.). Brooks/Cole Publishing Company. ISBN 0-534-19002-2.