Nil ideal
Encyclopedia
In mathematics
, more specifically ring theory
, an 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. Despite this example, the theory of nil ideals is most interesting in the case of noncommutative ring
s, where many problems still remain elusive—for instance, the Köthe conjecture.
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.
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.
, 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.
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 may generalize to right noetherian rings; a phenomenon known as Levitzky's theorem
, a particularly simple proof of which is due to Utumi.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, more specifically ring theory
Ring theory
In abstract algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers...
, an ideal
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
of a ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
is said to be a nil ideal if each of its elements is nilpotent. The nilradical of a commutative ring
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
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. Despite this example, the theory of nil ideals is most interesting in the case of noncommutative ring
Noncommutative ring
In mathematics, more specifically modern algebra and ring theory, a noncommutative ring is a ring whose multiplication is not commutative; that is, if R is a noncommutative ring, there exists a and b in R with a·b ≠ b·a, and conversely.Noncommutative rings are ubiquitous in mathematics, and occur...
s, where many problems still remain elusive—for instance, the Köthe conjecture.
Commutative rings
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.
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 2010.
Relation to nilpotent ideals
The notion of a nil ideal has a deep connection with that of a nilpotent idealNilpotent 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. By Ik, it is meant the additive subgroup generated by the set of all products of k elements in I...
, 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.
In a right artinian ring
Artinian ring
In abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals. They are also called Artin rings and are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are...
, any nil ideal is nilpotent. This is proven by observing that any nil ideal is contained in the Jacobson radical
Jacobson radical
In mathematics, more specifically ring theory, a branch of abstract algebra, the Jacobson radical of a ring R is an ideal which consists of those elements in R which annihilate all simple right R-modules. It happens that substituting "left" in place of "right" in the definition yields the same...
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
Levitzky's theorem
In mathematics, more specifically ring theory and the theory of nil ideals, Levitzky's theorem, named after Jacob Levitzki, states that in a right Noetherian ring, every nil one-sided ideal is necessarily nilpotent. Levitzky's theorem is one of the many results suggesting the veracity of the Köthe...
, a particularly simple proof of which is due to Utumi.