Nilpotent ideal

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.[2] 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.[3][4] The notion of a nilpotent ideal, although interesting in the case of commutative rings, is most interesting in the case of noncommutative rings.

Contents

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 reason than one. 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.[5]

In a right artinian ring, any nil ideal is nilpotent.[6] 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 may generalize to right noetherian rings; a phenomenon known as Levitzky's theorem, a particularly simple proof of which is due to Utumi.[7]

See also

Notes

  1. ^ Isaacs, p. 194
  2. ^ Isaacs, p. 194
  3. ^ Isaacs, Theorem 14.38, p. 210
  4. ^ Herstein, Theorem 1.4.5, p. 37
  5. ^ Isaacs, p. 194
  6. ^ Isaacs, Corollary 14.3, p. 195
  7. ^ Herstein, Theorem 1.4.5, p. 37

References

  • I.N. Herstein (1968). Noncommutative rings (1st edition ed.). The Mathematical Association of America. ISBN 0-88385-015-X. 
  • I. Martin Isaacs (1993). Algebra, a graduate course (1st edition ed.). Brooks/Cole Publishing Company. ISBN 0-534-19002-2. 

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