Nil Ideal - Relation To Nilpotent Ideals

Relation To Nilpotent 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. There are two main barriers for nil ideals to be nilpotent:

1. There need not be an upper bound on the exponent required to annihilate elements. Arbitrarily high exponents may be required.
2. The product of n nilpotent elements may be nonzero for arbitrarily high n.

Clearly both of these barriers must be avoided for a nil ideal to qualify as nilpotent.

In a right artinian ring, any nil ideal is nilpotent. 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 has been generalized to right noetherian rings; the result is known as Levitzky's theorem. A particularly simple proof due to Utumi can be found in (Herstein 1968, Theorem 1.4.5, p. 37).