Open problems in non-commutative algebra

Open problems in noncommutative algebra.

Everybody is invited to add, correct or edit (please try to provide references or attributions).

Infinite dimensional division algebras

• Kurosh problem for division algebras.
• The Kolchin-Plotkin problem: Let $D$ be a division ring. Can any unipotent subgroup $G\le G{L}_n(D)$ be simultaneously triangularized?
  • True for algebras over a field of characteristic zero, or characteristic sufficiently large compared to $n$ (Mochizuki). Even under these assumptions, the problem is still open for unipotent submonoids.
  • True for algebras over a field of characteristic 2 (Derakhshan-Wigner; by Sizer, nilpotency implies triangularizability)
• Is there a finitely generated infinite dimensional algebra over a field, which is a division algebra? (Latyshev; Ikeda - an equivalent formulation in terms of maximal left ideals in free algebras)
• Is it true that every division algebra is either locally PI or contains a noncommutative free subalgebra? (Makar-Limanov; Stafford)
• Let $D$ be a division algebra over a field $F$, which does not contain a noncommutative free subalgebra. Is it possible that $D{\otimes }_{F}K$ contains a noncommutative free subalgebra (for some field extension $K/F$)? (Makar-Limanov. When 'division algebra' is replaced by 'nil algebra', an example exists by Smoktunowicz)
• Let $D$ be a division algebra algebraic over a central subfield $F$. Must $M_n(D)$ be algebraic over $F$?
• Is it true that a division ring that is finitely generated over its center and left algebraic over some subfield is finite-dimensional over its center? (Bell, Drensky, Sharifi - here)
• Let $k$ be an algebraically closed field and let $A$ be a finitely generated Noetherian $k$-algebra, which is a domain that does not satisfy a polynomial identity. Is it possible for the quotient division algebra of $A$ to be left algebraic over some subfield? (Bell, Drensky, Sharifi - here)
• If a division ring $D$ is left algebraic over a subfield $K$ must $D$ also be right algebraic over $K$? (Bell, Drensky, Sharifi - here. The authors believe this problem was already posed before.)
• Suppose that $D_1\twoheadrightarrow D_2$ are Ore domains. If $Q(D_2)$ contains a free subalgebra, does $Q(D_1)$ contain a free subalgebra? (Greenfeld)
• Is there a finitely generated infinite dimensional Lie algebra whose universal enveloping algebra localized at its center is a division algebra? (After Shestakov-Zelmanov, who gave a specific candidate)

Noetherian rings

• Jacobson's conjecture: In a left and right Noetherian ring, is the intersection of all powers of the Jacobson radical zero? (Jacobson, 1956. Counterexamples for one sided Noetherian rings found by Herstein and Jategaonkar.)
• Herstein's conjecture: If $R$ is a left Noetherian ring, and $L\subseteq K\le R$ are left ideals such that $K$ is nil over $L$, then $K$ is nilpotent over $L$.
  • True for PI-rings, or for rings Artinian on one side (and not necessarily Noetherian on the other), by Herstein
  • If $L$ is a two-sided ideal then an affirmative answer follows from Levitski's theorem
• When does the Jacobson radical of a two-sided Noetherian ring $R$ satisfy the Artin-Rees property? In particular, does this occur if either $R/J(R)$ is Artinian or $R$ is prime? (Goodearl-Warfield)
• Can every right ideal in a simple Noetherian ring be generated by two elements?
  • Holds for the Weyl algebras (Stafford)
• Let $R$ be a finitely generated Noetherian algebra over a field $F$ of characteristic zero and le $K/F$ be a field extension. Must $R{\otimes }_{F}K$ be Noetherian?
  • True for PI-algebras (arbitrary characteristic; by Small)
  • True for $\mathbb{Z}$-graded algebras with finite dimensional homogeneous components (de Jong)
  • Counterexample in positive characteristic exist (Resco-Small) and in characteristic zero (Passman-Small, '23). It is open if a finitely presented example exists (Goodearl-Warfield)
  • True for countably generated algebras over uncountable algebraically closed fields (Bell)
  • There exist examples in arbitrary characteristic which are graded, Noetherian but non-Noetherian after an extension of the base field with a Noetherian commutative ring (Rogalski, "Generic noncommutative surfaces", Adv. Math. 2005)

• Must an affine Noetherian algebra be finitely presented? (Bergman, GK dim of factor rings; repeated by McConnell-Stafford. For PI algebras: Bell, 2004)
  • False for non-PI rings (Resco-Small, in characteristic p>0). To the best of our knowledge, this is still open for algebras over a field of characteristic zero (good candidate: the algebra from the aforementioned Passman-Small paper).
  • True for graded algebras (Lewin, Theorem 17)
  • True for PI algebras (Belov)

Primes in Noetherian rings

• Does a two-sided Noetherian ring satisfy DCC(primes)? Does every prime have finite height? Does every non-minimal prime contain a prime of height one? (Goodearl-Warfield)
  • True for PI-rings
• In a two-sided Noetherian ring, are all chains of ideals countable? In a finitely generated module over a Noetherian ring, are all chains of submodules countable? (Goodearl-Warfield)
  • True for commutative rings (Bass); false for one-sided Noetherian rings (Jategaonkar)
• In a two-sided Noetherian ring $R$, is the classical Krull dimension of $R[x]$ equal to the classical Krull dimension of $R$ plus one?
  • Well known for commutative rings

Dimensions in Noetherian rings

Global & projective dimension

• If $R$ is a two-sided Noetherian ring of finite global dimension and $R/J(R)$ is simple Artinian, is $R$ prime? (Goodearl-Warfield)
• If $R$ is a two-sided Noetherian ring of finite global dimension and $R/J(R)$ is a division ring, is $R$ a domain? (Goodearl-Warfield)
• Is the Krull dimension of $R$ bounded from above by its global dimension, for any two-sided Noetherian ring of finite global dimension? (Goodearl-Warfield)
  • For commutative rings, equality holds. Not true for one-sided Noetherian rings (Jategaonkar's example)

• Is the right global dimension of a two-sided Noetherian ring equal to the supremum of the projective dimensions of simple right modules? (Goodearl-Warfield)
  • True for commutative rings, or for rings finite module over their Noetherian centers. False for one-sided Noetherian rings (by Fields)
• If all simple right modules of a two-sided Noetherian ring have finite projective dimension, do all f.g. right modules have finite projective dinension? (Goodearl-Warfield)
  • True for commutative rings (Bass and Murthy) and module finite algebras over commutative Noetherian rings.

Krull dimension

• Do the right and left Krull dimensions of a two-sided Noetherian ring coincide? Of any Noetherian bimodule? (Goodearl-Warfield)

GK-dimension

• Is the GK-dimension exact for finitely generated over (affine) Noetherian algebras?
  • True if there is a filtration such that the associated graded is Noetherian (McConnell-Robson, 3.11)
  • True for affine Noetherian PI-algebras (Lenagan)
  • False for non-Noetherian algebras (even PI; Bergman)

Universal enveloping algebras

• Is there an infinite dimensional Lie algebra $L$ whose universal enveloping algebra is Noetherian? (Sierra-Walton: the universal enveloping algebra of the Witt algebra is not Noetherian; hence for $\mathbb{Z}$-graded simple Lie algebras of polynomial growth. For a group algebra counterpart of this question, see here.)
• Conjecture: the universal enveloping algebras of the Witt (and positive Witt) algebras satisfy ACC(ideals) (Petukhov-Sierra)
• Does the universal enveloping algebra of a loop algebra satisfy ACC(ideals)? (Sierra, Seattle '22)

Nil rings and radicals

• Does there exist a simple nil algebra over an uncountable field?
  • An example over a countable field exists, solving a question of Kaplansky (Smoktunowicz)
• Is there a finitely generated graded-nil ring (i.e. every homogeneous element is nilpotent), generated in degree 1, which contains a noncommutative free subalgebra? (Bell-Greenfeld. Examples not generated in degree 1 exist)
• Is there a graded, f.g. in degree 1 algebra all of whose homogeneous components satisfy the identity $x^n=0$ for some $n$?
  • Without the generation in degree 1 assumption -- examples exist

Kurosh type questions

• Does there exist an infinite dimensional finitely presented nil algebra? (Attributed to Ufnarovskii, repeated by many others)
• Is there a nil, non-nilpotent algebra whose adjoint group is finitely generated? (Amberg, Kazarin, Sysak)

Köthe type questions

• Köthe conjecture
• Let $R$ be a finitely generated nil algebra. Is $gr(R)$ Jacobson radical? (Riley, 2001)
  • True over uncountable fields (Alon Regev)
  • $gr(R)$ need not be nil even if $R$ is (Smoktunowicz)
• Suppose $R$ is a nil $F$-algebra and $K/F$ is a finite field extension. Must $R{\otimes }_{F}K$ be nil? Moreover, is this question equivalent to the Köthe conjecture? (Smoktunowicz)
• Let $R$ be a ring and deonte by $N$ the sum of nil ideals of $R$, and by $\tilde{N}$the sum of left nil ideals. Does $N=0$ imply $\bigcap _{i=1}^{\mathrm{\infty }}{\tilde{N}}^i=0$? Does $\bigcap _{i=1}^{\mathrm{\infty }}{\tilde{N}}^i=0$ imply $N=\tilde{N}$? (Rowen, 1989. Note that the conjunction of these questions implies an affirmative answer to the Köthe conjecture.)

Radicals of skew-polynomial and differential polynomial rings

• Let $F$ be a field of characteristic zero and let $R$ be an $F$-algebra and $\delta :R\to R$ a locally nilpotent derivation. Is $J(R[X;\delta ])=I[X]$ for some nil ideal $I◃R$?(Smoktunowicz, here)
  • True for fields of positive characteristic (Smoktunowicz)
• Let $R$ be an algebra without non-zero nil ideals, and let $\delta :R\to R$ be a derivation. Must $R[X;\delta ]$ be semiprimitive? (Smoktunowicz, here)

Prime ideals and prime spectra

• Does the universal enveloping algebra of the Witt algebra satisfy ACC(primes)? (Iyudu-Sierra: it does satisfy ACC(completely primes).)
• Is it true in any ring $R$ that for any pair of primes $P⊊Q◃R$ there exist primes: $P\subseteq P^{\prime }⊊Q^{\prime }\subseteq Q$ such that there is no intermediate prime between $P^{\prime }⊊Q^{\prime }$? (See here for some background and examples. True for PI-rings.)
• Characterize partially ordered sets which can be realized as $Spec(R)$ for some (not necessarily commutative) ring $R$. (See here for some background and examples.) Is the ordered set $([0,1],\le )$ isomorphic to some $(Spec(R),\subseteq )$?

Dixmier-Moeglin equivalence

• Does DME hold for (complex) affine Noetherian Hopf algebras of finite GK-dimension? (Bell, Seattle '22)
• Does DME hold for (complex) affine Noetherian twisted homogeneous coordinate rings? (Bell, Seattle '22)
• Does DME hold for (complex) affine prime Noetherian algebras of GK-dimension at most 3? (Bell, Seattle '22)

General structure theory

• Kurosh problem for simple algebras: Is there a finitely generated, infinite dimensional algebraic simple algebra? (Attributed by Smoktunowicz to Small)
• Is there an idempotent ring $R$ (not necessarily unital) which is not generated by one element as a bimodule over itself, namely, $R\ne RaR$ for any $a\in R$? (Monod, Ozawa, Thom)
  • True for semigroup algebras (Bergman/Smoktunowicz)
• Let $R$ be a principal ideal domain; if the units together with $0$ form a field $k$, is $R$ necessarily a polynomial ring over $k$? (A. Hausknecht, appears in Cohn's book)
• Is the notion of left integral extension transitive? (If every element of a ring $B$ is left integral over a subring $A$, then $B$ is called left integral over $A$. Appears in Cohn's book.)
• Which commutative rings occur as centers of Sylvester domains? Is the center of a Sylvester domain necessarily integrally closed? (Appears in Cohn's book)
• An $O$-ring is a unital ring in which every element other than the identity is a left and right zero divisor (example: a product of copies of the field with two elements). Is there a noncommutative $O$-ring?
  • An $O$-ring must be semiprime, but if it is prime, it is just $\mathbb{Z}/2\mathbb{Z}$. The question is equivalent to the question of whether any homomorphic image of an $O$-ring is again an $O$-ring. For resources and details, see here.
• Is there a finitely generated ring $R$ such that $R\cong R\times R$? (D. Osin, 2020, here. The group-theoretic counterpart has an affirmative answer, by Jones.)
• Is it true that every nilpotent matrix over a simple ring with unity can be presented as a commutator? (See here.)
• Is there a simple ring in which not every sum of commutators is a single commutator? In which not every sum of commutators is a sum of less than $n$ commutators, for given (or for all) $n$? (A positive answer to the latter would yield a counterexample to Question 6 here.)
• Is every prime ring an essential subring of a primitive ring? (Rowen, 1977, here. True by Goodearl's theorem for rings with a trivial center.)

Free algebras

• Let $R$ be a free $k$-algebra and $\hat{R}$ its completion by power series. Given $\alpha \in k$, denote by $C,C^{\prime }$ its centralizers in $R$,$\hat{R}$ respectively. Is $C^{\prime }$ the closure of $C$ in $\hat{R}$? (Bergman)
• Is every retract of a free algebra free? (A retract is a subring, which is also a homomorphic image of the containing ring under a homomorphism fixing the former. Attributed to Clark in Cohn's book)
• Is any endomorphism of a free algebra, carrying any primitive element to a primitive element necessarily an automorphism? (A primitive element is an element participating in a free basis. See here)
• Is the intersection of two retracts of a free algebra $R$ again a retract of $R$? (See here. Bergman proved the analogous result for free groups.)
• Let $R$ be a free algebra and $\hat{R}$ its power series completion. If an element of $R$ is a square in $\hat{R}$, is it associated (in $\hat{R}$) to the square of an element of $R$? (Two elements are associated if each one of them is a left and right product of the other by invertible elements. Bergman, appears in Cohn's book)
• Let $R$ be an algebra such that $M_n(R)$ contains a (noncommutative) free subalgebra. Must $R$ contain a free subalgebra? Same question for graded algebras. Seems unclear even for monomial algebras (Greenfeld)

Finite dimensional algebras

Central simple algebras

• Must a central division algebra of prime degree be cyclic?
• See this paper for a specialized list of problems on crossed product, exponent, the Brauer group, Brauer dimension and more.

Growth

Characterization and realization of growth functions

• Is there an asymptotic characterization of growth functions of finitely generated algebras?
  • There exists a characterization using discrete derivatives here (Bell-Zelmanov)
• Characterize growth rates of Lie algebras. Is any increasing exponentially bounded function equivalent to the growth of some finitely generated Lie algebra?
• Characterize growth rates of Hopf algebras (proposed by J. J. Zhang in Banff, 2022).

Growth of special classes of algebras

• Is the growth function of any algebra equivalent to the growth function of some primitive algebra? Or a nil algebra? (Zelmanov. Impossible if one restricts to graded primitive algebras.)

• Is there a finitely generated nil algebra with polynomially bounded growth over an arbitrary field?
  • Examples over countable fields exist, of GK-dim at most 3 (finite GK-dim by Lenagan-Smoktunowicz, and bound improved to 3 by Lenagan-Smoktunowicz-Young)
• Is there a finitely generated (even: graded, Noetherian, Artin-Schelter regular) domain of non-integral GK-dimension?
• Is there a finitely generated domain whose growth function is super-polynomial but asymptotically slower than $\exp (\sqrt{n})$?
  • For an example with growth $\exp (\sqrt{n})$, consider the universal enveloping algebra of any finitely generated Lie algebra of linear growth, by M. Smith.

• Is there an affine graded Noetherian algebra of super-polynomial growth? (Stephenson-Zhang, who proved it must be subexponential)

• Is the GK-dimension of the associated graded algebra of every Jacobi algebra with respect to the descending filtration of powers of the Jacobson radical, an integer? Brown--Wemyss, 2025

Dichotomy conjectures for low GK-dimension

• Let $R$ be a finitely generated prime Noetherian algebra of GK-dimension 2. Must $R$ be either primitive or PI? (Braun, Small)
• Let $R$ be a finitely generated prime algebra of quadratic growth. Must $R$ have bounded degrees of matrix images?
  • The answer is positive for monomial algebras; negative if growth restriction is relaxed to having GK-dim = 2 (Bell-Smoktunowicz). Unknown for finitely generated prime Noetherian algebras of GK-dim 2.
• Let $R$ be a finitely generated prime semiprimitive algebra of GK-dimension 2 (or: quadratic growth). Must $R$ be either primitive or PI? (Smoktunowicz, Vishne)
  • True for monomial algebras, without growth restrictions (Okn'inski)
• Let $R$ be a finitely generated algebra of quadratic growth. Must $R$ have finite classical Krull dimension?
  • False true for algebras of GK-dimension 2 (Bell)
  • True for graded algebras generated in degree 1, having quadratic growth (Greenfeld-Smoktunowicz-Leroy-Ziembowski)
  • False for graded (even monomial) algebras of GK-dimension 2 (Greenfeld)
• Is there a graded just infinite (also called projectively simple) algebra without a finitely generated module of GK-dimension 1? (Reichstein-Rogalski-Zhang. By Small-Zelmanov there exist graded, just infinite algebras without point modules). Related question: can a finitely generated infinite-dimensional nil graded algebra have a finitely generated infinite-dimensional module of finite width?
• Is there a non-PI finitely generated domain of GK-dimension 2 (or: less than 3) over a finite field? (Smoktunowicz)
• Conjecture: A non-PI, finitely generated domain of quadratic growth over an algebraically closed field of characteristic zero, which has a non-zero locally nilpotent derivation is Noetherian (Bell-Smoktunowicz, here).

Homological algebra

• Let $R$ be a connected, nonnegatively graded algebra (resp. Hopf algebra) over a field. Suppose either that $R$ is finitely presented or that $gl.dim(R)<\mathrm{\infty }$. Is it true that $R$ must have either subexponential or polynomial growth, or else contain a free subalgebra (resp. Hopf subalgebra) on two homogeneous generators? (Anick)
  • Finitely presented connected graded algebras with sufficiently sparse relators contain a free subalgebra (Smoktunowicz)
• Suppose $R$ is a connected graded algebra with polynomial growth and with $gl.dim(R)=d<\mathrm{\infty }$. Then the Hilbert series of $R$ is given by $\prod _{i=1}^d\frac{1}{1-z^{e_i}}$ for some positive integers $e_1,\dots ,e_d$ (Anick).
  • Holds for commutative algebras, enveloping algebras, monomial algebras and Noetherian PI-algebras. For details, see here.

Noncommutative algebra and algebraic geometry

Noncommutative projective geometry

• Classify noncommutative projective surfaces (Artin's proposed classification). Related problems: [1]
• Let $A$ be a connected graded finitely generated complex domain of GK-dimension 3 and suppose that $A$ has a nonzero locally nilpotent derivation. Then $Q_{gr}(A)\cong D[t,t^{-1};\sigma ]$ for some division algebra $D$ and $\sigma \in Aut(D)$. Can one describe the class of division algebras $D$ (or pairs $(D,\sigma )$) which can occur under these hypotheses? (Bell-Smoktunowicz, here)

Rings of differential operators

• Dixmier's conjecture
  • Stably equivalent to the Jacobian conjecture (Tsuchimoto; Belov-Kontsevich)
• The weak Gelfand-Kirillov conjecture: is the quotient division algebra of the universal enveloping algebra of an algebraic, finite-dimensional (complex) Lie algebra isomorphic, up to scalar extension, to the quotient division algebra of a Weyl algebr over a field of rational functions?
  • True, even without scalar extension, for solvable Lie algebras and semisimple of type $A_n$
  • Without scalar extension (original Gelfand-Kirillov conjecture) - false, by Alev-Ooms-Van den Bergh.
• Let $k$ be an algebraically closed field of characteristic 0 and let $A$ be a finitely generated $k$-algebra that is a domain of quadratic growth that is birationally isomorphic to a ring of differential operators on an affine curve over $k$. Does there exist a finitely generated subalgebra $B\subseteq Q(A)$ of quadratic growth that contains $A$ and has the property that the Weyl algebra is a subalgebra of $B$? (Bell-Smoktunowicz, here)
  • True if $A$ is a (non-PI) finitely generated domain having a non-zero locally nilpotent derivation (Bell-Smoktunowicz)
• Can one classify all pairs $(X,Y)$ of smooth affine complex curves such that the ring of differential operators $D(X)$ is isomorphic to a subalgebra of the quotient division ring of $D(Y)$? Conjecture: in such a case, the genus of $X$ is less than or equal to the genus of $Y$. (Bell-Smoktunowicz, here)
  • If $X$ is a smooth curve over an algebraically closed field of characteristic zero, then the quotient division ring of $D(X)$ always contains a copy of the Weyl algebra.

Polynomial identities

Representability

• Must a Noetherian PI-algebra over a field be representable? (Anan'in proved for affine Noetherian PI-algebras. In the non-affine case, seems to be open even for left and right Noetherian PI-algebras, and for Artinian PI-algebras.)
• Is every algebra over a field, that is a finitely generated module over its center, weakly representable (namely embeddable into a matrix ring over a commutative ring)? (Rowen, Small, J. Alg. 2015)
• Is every finitely presented PI-algebra over a field representable?
  • No (Irving)
• Suppose a commutative ring $C$ contains a field and $M$ is a finitely generated $C$-module. Is $En{d}_{C}M$ weakly representable? (Bergman, Isr. J. Math. 1970)

Group algebras

• Kaplansky's conjectures
  • A counterexample to the unit conjecture was announced by Giles Gardam (Feb '21)
• Let $G$ be a right-ordered group. Does its group algebra over a field $F[G]$ embed into a division ring?
• Let $G$ be a group. Is $\mathbb{Q}[G]$ semiprimitive?
  • $ℂ[G]$ is semiprimitive. For many refinements and variations on this question, as well as partial known results, see here (Passman).
• Let $G$ be a group whose group algebra $F[G]$ over some field is Noetherian. Does it follow that $G$ is virtually polycyclic?
  • The converse is well known. It is known that if the group algebra is Noetherian, then the underlying group is at least amenable. See discussion here.
  • Ivanov gave an example of a Noetherian group whose group ring over an arbitrary ring is not Noetherian, solving a question of P. Hall. See here.

Algebras defined by generators and relators

• Let $F$ be a field and let $f,g,h,u\in F[x]$ be irreducible polynomials of degree 2. The algebra:

$D=F⟨a,b,c,d⟩/⟨f(a),g(b),h(c),u(d),a+b+c+d⟩$ has quadratic growth (in fact, $2n+1$ monomials of degree $n$). Is $D$ a domain? (Bergman, the diamond lemma paper.) Note that if true, this might give an example of a non-PI domain of GK-dimension 2 over a finite field.

Nonassociative structures

• See the Dniester "notebook".

Poisson algebras

• Let $A$ be a simple Poisson algebra. To which extent does the Lie algebra $\{A,A\}$ determine $A$?