1 · The essentials
The manuscript describes a number-like system in which division by any nonzero element is possible, but changing the order of multiplication can change the answer. Inside it is a core of elements that can be multiplied in either order with every element without changing the result. Every element of the system satisfies a polynomial equation with coefficients from this core. Just two elements, together with the core, generate the whole system using addition, subtraction, and multiplication. Yet no finite list of elements can express everything by taking sums of core multiples. The manuscript presents this combination as a negative answer to Kurosh’s division-ring problem. The main result has not been formalized in Lean and has not been formally verified.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine a workshop where two tools, together with a stock of basic supplies, can produce every item. Having only two starting tools does not mean that every item can be assembled as a simple combination of a fixed finite catalogue of parts. The image captures the distinction between generating everything by repeated operations and describing everything by finite sums of basic pieces.
Why it matters
The result separates two kinds of finiteness: each element satisfying a polynomial equation, and finitely generated parts having finite dimension over the center. The manuscript’s construction shows that the first need not force the second for division rings, even when the whole ring is generated by two elements over its center.
2 · Going further
Kurosh’s division-ring problem, as described in the repository summary, asks about local finiteness for division rings algebraic over their centers. Here the center of a division ring consists of the elements that commute with every element of . Being algebraic over means that each satisfies a nonzero polynomial equation with coefficients in . Local finiteness asks whether every subalgebra generated by finitely many elements over has finite dimension over .
The manuscript states that it constructs a countable division ring of characteristic zero, algebraic over its center , with as an algebra but . These properties give the stated negative answer: the entire ring is already a finitely generated subalgebra over its center, yet it has infinite dimension over that center. Algebra generation allows repeated products of the generators; finite-dimensionality would require a finite list whose -linear combinations cover the ring.
The supplied abstract announces the construction but does not describe its mechanism or the steps of the argument. Thus the available material supports an explanation of how the stated properties yield a counterexample, not an account of how the ring is built. Only the main manuscript is supplied; no companion manuscripts or relationships to them are described.
Useful notions
- Polynomial equations
- Rings and division rings
- Centers and commuting elements
- Vector-space dimension
- Generation of an algebra
What was known before / what is new
Before
The source identifies Kurosh’s division-ring problem as a question about local finiteness for division rings algebraic over their centers. It gives no earlier partial results or historical account beyond naming the problem.
New
The manuscript claims a countable characteristic-zero division ring that is algebraic over its center and generated by two elements as an algebra over that center, while remaining infinite-dimensional over it. The repository summary and abstract present this construction as a negative answer to Kurosh’s division-ring problem.
Limits and assumptions
The stated construction is countable, has characteristic zero, and is algebraic over its center; its two-generator and infinite-dimensional properties are both relative to that center. The supplied abstract contains neither the construction details nor the argument needed to assess them. The main result is not formalized in Lean and has not been formally verified. The source does not identify further open questions or state what happens under additional restrictions.
Mini glossary
- Division ring —
- A ring in which every nonzero element has a multiplicative inverse, without requiring multiplication to commute.
- Center —
- The elements of a ring that commute with every element of the ring.
- Algebraic over the center —
- Having every element satisfy a nonzero polynomial equation with coefficients in the center.
- Generated as an algebra —
- Obtainable from the specified generators and coefficient field by finite applications of addition, subtraction, and multiplication.
- Local finiteness —
- The property that every subalgebra generated by finitely many elements over the center has finite dimension over that center.
- Characteristic zero —
- The property that no positive number of copies of the multiplicative identity adds up to zero.
Keywords : Division rings · Kurosh’s problem · Local finiteness
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
Constructs a countable characteristic-zero division ring that is algebraic over its center and generated by two elements over that center, but has infinite dimension over it. This answers Kurosh's division-ring problem on local finiteness negatively.
To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.
Manuscripts in this family
A Counterexample to Kurosh’s Division-Ring Problem
23/09/2026We construct a countable division ring of characteristic zero that is algebraic over its center, generated by two elements as an algebra over that center, and infinite-dimensional over it. This gives a negative answer to the Kurosh problem for division rings.