Nilpotent ideal

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 Ik = 0.[1] By Ik, 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

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

Notes

  1. 1 2 3 Isaacs, p. 194.
  2. ↑ Isaacs, Theorem 14.38, p. 210
  3. 1 2 Herstein, Theorem 1.4.5, p. 37.
  4. ↑ Isaacs, Corollary 14.3, p. 195

References

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