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. Commutative rings In commutative rings, the nil ideals are better understood than in noncommutative rings, primarily because in commutative rings, products involving nilpotent elements and sums of nilpotent elements are both nilpotent. This is because if a and b are nilpotent elements of R with an=0 and bm=0, and r is any element of R, then (a·r)n = an·rn = 0, and by the binomial theorem, (a+b)m+n=0. Therefore, the set of all nilpotent elements forms an ideal known as the nilradical of a ring. Because the nilradical contains every nilpotent element, an ideal of a commutative ring is nil if and only if it is a subset of the nilradical, and so the nilradical is maximal among nil ideals. Furthermore, for any nilpotent element a of a commutative ring R, the ideal aR is nil. For a noncommutative ring however, it is not in general true that the set of nilpotent elements forms an ideal, or that a·R is a nil (one-sided) ideal, even if a is nilpotent. Noncommutative rings 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 2022. Relation to nilpotent ideals The notion of a nil ideal has a