Math 2.0

Associative rings and algebras Chain, growth and finiteness conditions Problem

Kurosh problem

Builds on Associative algebraFieldAlgebraic elementNil idealGolod–Shafarevich theoremPolynomial identity ringBurnside problemDivision ring

Problem

Let kk be a field and let AA be an associative algebra over kk that is generated by finitely many elements. Assume that AA is algebraic: every element of AA is algebraic over kk, that is, a root of a nonzero polynomial with coefficients in kk. Must AA be finite-dimensional over kk?

Remarks

The problem is a ring-theoretic analogue of the Burnside problem for periodic groups. An important special case asks whether a finitely generated nil algebra, in which every element is nilpotent, must be nilpotent and hence finite-dimensional.

The answer is negative in general. Golod used the Golod–Shafarevich theorem in 1964 to construct a finitely generated infinite-dimensional nil algebra. The same construction gave infinite finitely generated periodic groups, a negative answer to the general Burnside problem.

The answer is positive for algebras that satisfy a polynomial identity: a finitely generated algebraic polynomial identity algebra is finite-dimensional. In particular it is positive when the elements of AA are algebraic of bounded degree.

A variant asks the same question for division rings that are algebraic over their centre and finitely generated over it.

Reading path

To understand this page from the ground up, read these 20 pages in order. The foundations come first.

Show the path as a graph
Algebraic element, concept, no evidence levelAlgebraic elementconceptAssociative algebra, concept, no evidence levelAssociative algebraconceptDivision ring, concept, no evidence levelDivision ringconceptExponent of a group, concept, no evidence levelExponent of a groupconceptField, concept, no evidence levelFieldconceptFinitely generated group, concept, no evidence levelFinitely generated gr…conceptFormal power series, concept, no evidence levelFormal power seriesconceptFree algebra, concept, no evidence levelFree algebraconceptFree group, concept, no evidence levelFree groupconceptGraded ring, concept, no evidence levelGraded ringconceptHilbert class field, concept, no evidence levelHilbert class fieldconceptNil ideal, concept, no evidence levelNil idealconceptOrder of an element, concept, no evidence levelOrder of an elementconceptP-group, concept, no evidence levelP-groupconceptPolynomial identity ring, concept, no evidence levelPolynomial identity r…conceptTorsion group, concept, no evidence levelTorsion groupconceptTwo-sided ideal, concept, no evidence levelTwo-sided idealconceptBurnside problem, problem, no evidence levelBurnside problemproblemGolod–Shafarevich theorem, statement, no evidence levelGolod–Shafarevich the…statementKurosh problem, problem, no evidence levelKurosh problemproblem
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Algebraic element concept
  2. Associative algebra concept
  3. Division ring concept
  4. Exponent of a group concept
  5. Field concept
  6. Finitely generated group concept
  7. Formal power series concept
  8. Free algebra concept
  9. Free group concept
  10. Graded ring concept
  11. Hilbert class field concept
  12. Nil ideal concept
  13. Order of an element concept
  14. P-group concept
  15. Polynomial identity ring concept
  16. Torsion group concept
  17. Two-sided ideal concept
  18. Burnside problem problem
  19. Golod–Shafarevich theorem statement
  20. Kurosh problem problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Kurosh_problem
About this page
TagB1Q5F5VNSK
URIm2://B1Q5F5VNSK
Kindproblem
WritingW0 stub
MSC16N40, 16P90, 16R10
StatusThis copy of the site has no build data. Run m2 status B1Q5F5VNSK.
AboutAssociative algebra
UsesAssociative algebra, Field, Algebraic element, Nil ideal, Golod–Shafarevich theorem, Polynomial identity ring, Burnside problem, Division ring
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Changed2 November 2026

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