In mathematics, the Kurosh problem is one general problem, and several more special questions, in

ring theory In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their r ...

. The general problem is known to have a negative solution, since one of the special cases has been shown to have

counterexample A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "John Smith is not a lazy student" is ...

s. These matters were brought up by

Aleksandr Gennadievich Kurosh Aleksandr Gennadyevich Kurosh (russian: Алекса́ндр Генна́диевич Ку́рош; January 19, 1908 – May 18, 1971) was a Soviet mathematician, known for his work in abstract algebra. He is credited with writing ''The Theory o ...

as analogues of the

Burnside problem The Burnside problem asks whether a finitely generated group in which every element has finite order must necessarily be a finite group. It was posed by William Burnside in 1902, making it one of the oldest questions in group theory and was influ ...

group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...

. 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 In mathematics, specifically in ring theory, a nilpotent algebra over a commutative ring is an algebra over a commutative ring, in which for some positive integer ''n'' every product containing at least ''n'' elements of the algebra is zero. The co ...

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 ''I'p'', the localization of ''I'' at ''p'', is a nilpotent ideal in ''A'p''. In non-commutative ...

. For PI-algebras the Kurosh problem has a positive solution.

Golod Golod, also transliterated Holod from uk, Голод, is an East Slavic surname meaning ''hunger''. Notable people with the surname include: *Alexander Golod, Ukrainian pyramidologist * Evgeny Golod (1935–2018), Russian mathematician **Golod–S ...

showed a counterexample to that case, as an application of the

Golod–Shafarevich theorem In mathematics, the Golod–Shafarevich theorem was proved in 1964 by Evgeny Golod and Igor Shafarevich. It is a result in non-commutative homological algebra which solves the class field tower problem, by showing that class field towers can be ...

. The Kurosh problem on group algebras concerns the

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 R /math> to R, defin ...

''I''. If ''I'' is a

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., p. 194 The nilradical of a commutative ring is an example of a nil ideal; in fact, it i ...

, is the group algebra locally nilpotent? There is an important problem which is often referred as the Kurosh's problem on

division ring In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element ...

s. The problem asks whether there exists an algebraic (over the

center Center or centre may refer to: Mathematics *Center (geometry), the middle of an object * Center (algebra), used in various contexts ** Center (group theory) ** Center (ring theory) * Graph center, the set of all vertices of minimum eccentricity ...

) division ring which is not locally finite. This problem has not been solved until now.

References

*Vesselin S. Drensky, Edward Formanek (2004), ''Polynomial Identity Rings'', p. 89.
open problems in the theory of infinite dimensional algebras''
(2007). E. Zelmanov. Ring theory Unsolved problems in mathematics {{Abstract-algebra-stub