Nil ideal

In mathematics, more specifically ring theory, a left, right or two-sided ideal of a ring is said to be a nil ideal if each of its elements is nilpotent.

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.

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