The nilradical of a commutative ring is an example of a nil ideal; in fact, it is the ideal of the ring maximal with respect to the property of being nil. Unfortunately the set of nil elements does not always form an ideal for noncommutative rings. Nil ideals are still associated with interesting open questions, especially the unsolved Köthe conjecture.
In a commutative ring, the set of all nilpotent elements forms an ideal known as the nilradical of the ring. Therefore, an ideal of a commutative ring is nil if, and only if, it is a subset of the nilradical; that is, the nilradical is the ideal maximal with respect to the property that each of its elements is nilpotent.
In commutative rings, the nil ideals are more well-understood compared to the case of noncommutative rings. This is primarily because the commutativity assumption ensures that the product of two nilpotent elements is again nilpotent. For instance, if a is a nilpotent element of a commutative ring R, a·R is an ideal that is in fact nil. This is because any element of the principal ideal generated by a is of the form a·r for r in R, and if an = 0, (a·r)n = an·rn = 0. It is not in general true however, that a·R is a nil (one-sided) ideal in a noncommutative ring, even if a is nilpotent.
The theory of nil ideals is of major importance in noncommutative ring theory. In particular, through the understanding of nil rings—rings whose every element is nilpotent—one may obtain a much better understanding of more general rings.
In the case of commutative rings, there is always a maximal nil ideal: the nilradical of the ring. The existence of such a maximal nil ideal in the case of noncommutative rings is guaranteed by the fact that the sum of nil ideals is again nil. However, the truth of the assertion that the sum of two left nil ideals is again a left nil ideal remains elusive; it is an open problem known as the Köthe conjecture. The Köthe conjecture was first posed in 1930 and yet remains unresolved as of 2010.
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:
- There need not be an upper bound on the exponent required to annihilate elements. Arbitrarily high exponents may be required.
- 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).
- 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
- Smoktunowicz, Agata (2006), "Some results in noncommutative ring theory", International Congress of Mathematicians, Vol. II, Zürich: European Mathematical Society, pp. 259–269, ISBN 978-3-03719-022-7, MR 2275597, retrieved 2009-08-19