Kurosh problem
Encyclopedia
In mathematics
, the Kurosh problem is one general problem, and several more special questions, in ring theory
. The general problem is known to have a negative solution, since one of the special cases has been shown to have counterexample
s. These matters were brought up by Aleksandr Gennadievich Kurosh
as analogues of the Burnside problem in group theory
.
Kurosh asked whether there can be a finitely-generated infinite-dimensional algebraic algebra (the problem being to show this cannot happen). A special case is whether or not every nil algebra is locally nilpotent
.
Golod showed a counterexample to that case, as an application of the Golod-Shafarevich lemma.
The Kurosh problem on group algebra
s concerns the augmentation ideal
I. If I is a nil ideal
, is the group algebra locally nilpotent?
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...
, the Kurosh problem is one general problem, and several more special questions, in 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...
. The general problem is known to have a negative solution, since one of the special cases has been shown to have counterexample
Counterexample
In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule. For example, consider the proposition "all students are lazy"....
s. These matters were brought up by Aleksandr Gennadievich Kurosh
Aleksandr Gennadievich Kurosh
Aleksandr Gennadievich Kurosh was a Soviet mathematician, known for his work in abstract algebra. He is credited with writing the first modern and high-level text on group theory, his The Theory of Groups published in 1944.He was born in Yartsevo near Smolensk, and died in Moscow. He received his...
as analogues of the Burnside problem in group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
.
Kurosh asked whether there can be a finitely-generated infinite-dimensional algebraic algebra (the problem being to show this cannot happen). A special case is whether or not every nil algebra is locally nilpotent
Locally nilpotent
In the mathematical field of commutative algebra, an ideal I in a commutative ring A is locally nilpotent at a prime ideal p if the ideal is nilpotent in a Zariski open neighborhood of p...
.
Golod showed a counterexample to that case, as an application of the Golod-Shafarevich lemma.
The Kurosh problem on group algebra
Group algebra
In mathematics, the group algebra is any of various constructions to assign to a locally compact group an operator algebra , such that representations of the algebra are related to representations of the group...
s concerns the augmentation ideal
Augmentation ideal
In algebra, an augmentation ideal is an ideal that can be defined in any group ring. If G is a group and R a commutative ring, there is a ring homomorphism \varepsilon, called the augmentation map, from the group ring...
I. If I is a nil ideal
Nil ideal
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...
, is the group algebra locally nilpotent?