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

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* Kurosh problem for division algebras.
* The Kolchin-Plotkin problem: Let  D {\displaystyle D} be a division ring. Can any unipotent subgroup  G ≤ G L n ( D ) {\displaystyle G\leq GL_{n}(D)} be simultaneously triangularized?
* True for algebras over a field of characteristic zero, or characteristic sufficiently large compared to  n {\displaystyle 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 {\displaystyle D} be a division algebra over a field  F {\displaystyle F} , which does not contain a noncommutative free subalgebra. Is it possible that  D ⊗ F K {\displaystyle D\otimes _{F}K} contains a noncommutative free subalgebra (for some field extension  K / F {\displaystyle K/F} )? (Makar-Limanov. When 'division algebra' is replaced by 'nil algebra', an example exists by Smoktunowicz)
* Let  D {\displaystyle D} be a division algebra algebraic over a central subfield  F {\displaystyle F} . Must  M n ( D ) {\displaystyle M_{n}(D)} be algebraic over  F {\displaystyle 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 {\displaystyle k} be an algebraically closed field and let  A {\displaystyle A} be a finitely generated Noetherian  k {\displaystyle k} -algebra, which is a domain that does not satisfy a polynomial identity. Is it possible for the quotient division algebra of  A {\displaystyle A} to be left algebraic over some subfield? (Bell, Drensky, Sharifi - here)
* If a division ring  D {\displaystyle D} is left algebraic over a subfield  K {\displaystyle K} must  D {\displaystyle D} also be right algebraic over  K {\displaystyle K} ? (Bell, Drensky, Sharifi - here. The authors believe this problem was already posed before.)
* Suppose that  D 1 ↠ D 2 {\displaystyle D_{1}\twoheadrightarrow D_{2}} are Ore domains. If  Q ( D 2 ) {\displaystyle Q(D_{2})} contains a free subalgebra, does  Q ( D 1 ) {\displaystyle 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

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* 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 {\displaystyle R} is a left Noetherian ring, and  L ⊆ K ≤ R {\displaystyle L\subseteq K\leq R} are left ideals such that  K {\displaystyle K} is nil over  L {\displaystyle L} , then  K {\displaystyle K} is nilpotent over  L {\displaystyle L} .
* True for PI-rings, or for rings Artinian on one side (and not necessarily Noetherian on the other), by Herstein
* If  L {\displaystyle 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 {\displaystyle R} satisfy the Artin-Rees property? In particular, does this occur if either  R / J ( R ) {\displaystyle R/J(R)} is Artinian or  R {\displaystyle 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 {\displaystyle R} be a finitely generated Noetherian algebra over a field  F {\displaystyle F} of characteristic zero and le  K / F {\displaystyle K/F} be a field extension. Must  R ⊗ F K {\displaystyle R\otimes _{F}K} be Noetherian?
* True for PI-algebras (arbitrary characteristic; by Small)
* True for  Z {\displaystyle \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

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* 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 {\displaystyle R} , is the classical Krull dimension of  R [ x ] {\displaystyle R[x]} equal to the classical Krull dimension of  R {\displaystyle R} plus one?
* Well known for commutative rings

### Dimensions in Noetherian rings

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#### Global & projective dimension

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* If  R {\displaystyle R} is a two-sided Noetherian ring of finite global dimension and  R / J ( R ) {\displaystyle R/J(R)} is simple Artinian, is  R {\displaystyle R} prime? (Goodearl-Warfield)
* If  R {\displaystyle R} is a two-sided Noetherian ring of finite global dimension and  R / J ( R ) {\displaystyle R/J(R)} is a division ring, is  R {\displaystyle R} a domain? (Goodearl-Warfield)
* Is the Krull dimension of  R {\displaystyle 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

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

#### GK-dimension

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* 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

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* Is there an infinite dimensional Lie algebra  L {\displaystyle L} whose universal enveloping algebra is Noetherian? (Sierra-Walton: the universal enveloping algebra of the Witt algebra is not Noetherian; hence for  Z {\displaystyle \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

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* 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 {\displaystyle x^{n}=0} for some  n {\displaystyle n} ?
* Without the generation in degree 1 assumption -- examples exist

### Kurosh type questions

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* 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

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* Köthe conjecture
* Let  R {\displaystyle R} be a finitely generated nil algebra. Is  g r ( R ) {\displaystyle gr(R)} Jacobson radical? (Riley, 2001)
* True over uncountable fields (Alon Regev)
* g r ( R ) {\displaystyle gr(R)} need not be nil even if  R {\displaystyle R} is (Smoktunowicz)
* Suppose  R {\displaystyle R} is a nil  F {\displaystyle F} -algebra and  K / F {\displaystyle K/F} is a finite field extension. Must  R ⊗ F K {\displaystyle R\otimes _{F}K} be nil? Moreover, is this question equivalent to the Köthe conjecture? (Smoktunowicz)
* Let  R {\displaystyle R} be a ring and deonte by  N {\displaystyle N} the sum of nil ideals of  R {\displaystyle R} , and by  N ~ {\displaystyle {\widetilde {N}}} the sum of left nil ideals. Does  N = 0 {\displaystyle N=0} imply  ⋂ i = 1 ∞ N ~ i = 0 {\displaystyle \bigcap _{i=1}^{\infty }{\widetilde {N}}^{i}=0} ? Does  ⋂ i = 1 ∞ N ~ i = 0 {\displaystyle \bigcap _{i=1}^{\infty }{\widetilde {N}}^{i}=0} imply  N = N ~ {\displaystyle N={\widetilde {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

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

## Prime ideals and prime spectra

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* 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 {\displaystyle R} that for any pair of primes  P ⊊ Q ◃ R {\displaystyle P\subsetneq Q\triangleleft R} there exist primes:  P ⊆ P ′ ⊊ Q ′ ⊆ Q {\displaystyle P\subseteq P'\subsetneq Q'\subseteq Q} such that there is no intermediate prime between  P ′ ⊊ Q ′ {\displaystyle P'\subsetneq Q'} ? (See here for some background and examples. True for PI-rings.)
* Characterize partially ordered sets which can be realized as  S p e c ( R ) {\displaystyle Spec(R)} for some (not necessarily commutative) ring  R {\displaystyle R} . (See here for some background and examples.) Is the ordered set  ( [ 0 , 1 ] , ≤ ) {\displaystyle ([0,1],\leq )} isomorphic to some  ( S p e c ( R ) , ⊆ ) {\displaystyle (Spec(R),\subseteq )} ?

### Dixmier-Moeglin equivalence

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* 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

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* 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 {\displaystyle R} (not necessarily unital) which is not generated by one element as a bimodule over itself, namely,  R ≠ R a R {\displaystyle R\neq RaR} for any  a ∈ R {\displaystyle a\in R} ? (Monod, Ozawa, Thom)
* True for semigroup algebras (Bergman/Smoktunowicz)
* Let  R {\displaystyle R} be a principal ideal domain; if the units together with  0 {\displaystyle 0} form a field  k {\displaystyle k} , is  R {\displaystyle R} necessarily a polynomial ring over  k {\displaystyle k} ? (A. Hausknecht, appears in Cohn's book)
* Is the notion of left integral extension transitive? (If every element of a ring  B {\displaystyle B} is left integral over a subring  A {\displaystyle A} , then  B {\displaystyle B} is called left integral over  A {\displaystyle 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 {\displaystyle 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 {\displaystyle O} -ring?
* An  O {\displaystyle O} -ring must be semiprime, but if it is prime, it is just  Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } . The question is equivalent to the question of whether any homomorphic image of an  O {\displaystyle O} -ring is again an  O {\displaystyle O} -ring. For resources and details, see here.
* Is there a finitely generated ring  R {\displaystyle R} such that  R ≅ R × R {\displaystyle 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 {\displaystyle n} commutators, for given (or for all)  n {\displaystyle 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.)

## Metric noncommutative algebra

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* Is there a non linear sofic group (equivalently, group algebra)?
* Is every division algebra linear sofic? (Elek, Arzhantseva-Paunescu). Specifically, is Makar-Limanov's algebraically closed division ring linear sofic? (Greenfeld)
* Does every noncommutative polynomial equation have approximate solutions in terms of rank? False for systems of more than one equations (even if the quotient algebra is non-zero). Equivalently, does every one relator algebra have a homomorphism to a metric ultrproduct of matrix rings? If Makar-Limanov's division ring is linear sofic then yes (for characteristic zero).
* If two matrices almost commute in the rank distance, are they closed (in rank) to genuinely commuting matrices?
* Bartholdi-Kielak proved that for a torsion free group, its group algebra is an Ore domain if and only if the group is amenable. Is there an analogous result for Lie algebras? We can define amenability through the universal enveloping algebra.

## Free algebras

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* Let  R {\displaystyle R} be a free  k {\displaystyle k} -algebra and  R ^ {\displaystyle {\widehat {R}}} its completion by power series. Given  α ∈ k {\displaystyle \alpha \in k} , denote by  C , C ′ {\displaystyle C,C'} its centralizers in  R {\displaystyle R} , R ^ {\displaystyle {\widehat {R}}} respectively. Is  C ′ {\displaystyle C'} the closure of  C {\displaystyle C} in  R ^ {\displaystyle {\widehat {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 {\displaystyle R} again a retract of  R {\displaystyle R} ? (See here. Bergman proved the analogous result for free groups.)
* Let  R {\displaystyle R} be a free algebra and  R ^ {\displaystyle {\widehat {R}}} its power series completion. If an element of  R {\displaystyle R} is a square in  R ^ {\displaystyle {\widehat {R}}} , is it associated (in  R ^ {\displaystyle {\widehat {R}}} ) to the square of an element of  R {\displaystyle 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 {\displaystyle R} be an algebra such that  M n ( R ) {\displaystyle M_{n}(R)} contains a (noncommutative) free subalgebra. Must  R {\displaystyle R} contain a free subalgebra? Same question for graded algebras (Greenfeld)

## Finite dimensional algebras

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### Central simple algebras

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* 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

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### Characterization and realization of growth functions

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* 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

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* 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 ⁡ ( n ) {\displaystyle \exp({\sqrt {n}})} ?
* For an example with growth  exp ⁡ ( n ) {\displaystyle \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

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* Let  R {\displaystyle R} be a finitely generated prime Noetherian algebra of GK-dimension 2. Must  R {\displaystyle R} be either primitive or PI? (Braun, Small)
* Let  R {\displaystyle R} be a finitely generated prime algebra of quadratic growth. Must  R {\displaystyle 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 {\displaystyle R} be a finitely generated prime semiprimitive algebra of GK-dimension 2 (or: quadratic growth). Must  R {\displaystyle R} be either primitive or PI? (Smoktunowicz, Vishne)
* True for monomial algebras, without growth restrictions (Okn'inski)
* Let  R {\displaystyle R} be a finitely generated algebra of quadratic growth. Must  R {\displaystyle R} have finite classical Krull dimension?
* False 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

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* Let  R {\displaystyle R} be a connected, nonnegatively graded algebra (resp. Hopf algebra) over a field. Suppose either that  R {\displaystyle R} is finitely presented or that  g l . d i m ( R ) < ∞ {\displaystyle gl.dim(R)<\infty } . Is it true that  R {\displaystyle 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 {\displaystyle R} is a connected graded algebra with polynomial growth and with  g l . d i m ( R ) = d < ∞ {\displaystyle gl.dim(R)=d<\infty } . Then the Hilbert series of  R {\displaystyle R} is given by  ∏ i = 1 d 1 1 − z e i {\displaystyle \prod _{i=1}^{d}{\frac {1}{1-z^{e_{i}}}}} for some positive integers  e 1 , … , e d {\displaystyle 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

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### Noncommutative projective geometry

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

### Rings of differential operators

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* 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 {\displaystyle A_{n}}
* Without scalar extension (original Gelfand-Kirillov conjecture) - false, by Alev-Ooms-Van den Bergh.
* Let  k {\displaystyle k} be an algebraically closed field of characteristic 0 and let  A {\displaystyle A} be a finitely generated  k {\displaystyle 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 {\displaystyle k} . Does there exist a finitely generated subalgebra  B ⊆ Q ( A ) {\displaystyle B\subseteq Q(A)} of quadratic growth that contains  A {\displaystyle A} and has the property that the Weyl algebra is a subalgebra of  B {\displaystyle B} ? (Bell-Smoktunowicz, here)
* True if  A {\displaystyle A} is a (non-PI) finitely generated domain having a non-zero locally nilpotent derivation (Bell-Smoktunowicz)
* Can one classify all pairs  ( X , Y ) {\displaystyle (X,Y)} of smooth affine complex curves such that the ring of differential operators  D ( X ) {\displaystyle D(X)} is isomorphic to a subalgebra of the quotient division ring of  D ( Y ) {\displaystyle D(Y)} ? Conjecture: in such a case, the genus of  X {\displaystyle X} is less than or equal to the genus of  Y {\displaystyle Y} . (Bell-Smoktunowicz, here)
* If  X {\displaystyle X} is a smooth curve over an algebraically closed field of characteristic zero, then the quotient division ring of  D ( X ) {\displaystyle D(X)} always contains a copy of the Weyl algebra.

## Polynomial identities

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### Representability

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* 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 {\displaystyle C} contains a field and  M {\displaystyle M} is a finitely generated  C {\displaystyle C} -module. Is  E n d C M {\displaystyle End_{C}M} weakly representable? (Bergman, Isr. J. Math. 1970)

## Group algebras

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* Kaplansky's conjectures
* A counterexample to the unit conjecture was announced by Giles Gardam (Feb '21)
* Let  G {\displaystyle G} be a right-ordered group. Does its group algebra over a field  F [ G ] {\displaystyle F[G]} embed into a division ring?
* Let  G {\displaystyle G} be a group. Is  Q [ G ] {\displaystyle \mathbb {Q} [G]} semiprimitive?
* C [ G ] {\displaystyle \mathbb {C} [G]} is semiprimitive. For many refinements and variations on this question, as well as partial known results, see here (Passman).
* Let  G {\displaystyle G} be a group whose group algebra  F [ G ] {\displaystyle F[G]} over some field is Noetherian. Does it follow that  G {\displaystyle 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

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* Let  F {\displaystyle F} be a field and let  f , g , h , u ∈ F [ x ] {\displaystyle 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 ⟩ {\displaystyle D=F\langle a,b,c,d\rangle /\langle f(a),g(b),h(c),u(d),a+b+c+d\rangle } has quadratic growth (in fact,  2 n + 1 {\displaystyle 2n+1} monomials of degree  n {\displaystyle n} ). Is  D {\displaystyle 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

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* See the Dniester "notebook".

### Poisson algebras

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