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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 $$G\leq GL_n(D)$$ 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 triangularizbility)
 * 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 divsion 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 $$L\subseteq K\leq R$$ 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 characeristic 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

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 $$\widetilde{N}$$the sum of left nil ideals. Does $$N=0$$ imply $$\bigcap_{i=1}^{\infty}\widetilde{N}^i=0$$? Does $$\bigcap_{i=1}^{\infty}\widetilde{N}^i=0$$ imply $$N=\widetilde{N}$$? (Rowen, 1989. Note that the conjunction of these questions implies an affirmative answer to the Köthe conjecutre.)

Radicals of skew-polynomial and differential polynomial rings

 * Let $$F$$ be a field of characteristic zero and let $$R$$ be an $$F$$-algebra and $$\delta:R\rightarrow R$$ a locally nilpotent derivation. Is $$J(R[X;\delta])=I[X]$$ for some nil ideal $$I\triangleleft R$$?(Smoktunowicz, here)
 * True for fields of positive characteristic (Smoktunowicz)
 * Let $$R$$ be an algebra without non-zero nil ideals, and let $$\delta:R\rightarrow R$$ 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 $$P\subsetneq Q\triangleleft R$$ there exist primes: $$P\subseteq P'\subsetneq Q'\subseteq Q$$ such that there is no intermediate prime between $$P'\subsetneq Q'$$? (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 oredered set $$([0,1],\leq)$$ 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 homoegneous 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, $$R\neq RaR$$ 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 thre 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 genrated 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 $$\widehat{R}$$ its completion by power series. Given $$\alpha\in k$$, denote by $$C,C'$$ its centralizers in $$R$$,$$\widehat{R}$$ respectively. Is $$C'$$ the closure of $$C$$ in $$\widehat{R}$$? (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 $$R$$ again a retract of $$R$$? (See here. Bergman proved the analogous result for free groups.)
 * Let $$R$$ be a free algebra and $$\widehat{R}$$ its power series completion. If an element of $$R$$ is a square in $$\widehat{R}$$, is it associated (in $$\widehat{R}$$) 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 invertiable 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)

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.

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 equvalent 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 asyptotically 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)

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 ifeld 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 $$gl.dim(R)<\infty$$. 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 $$gl.dim(R)=d<\infty$$. 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 projective geometry

 * Classify noncommutative projective surfaces (Artin's proposed classification). Related problems:
 * 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 hypothese? (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 $$D(Y)$$? Conjuecture: 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.

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 $$End_CM$$ 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?
 * $$\mathbb{C}[G]$$ 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 undrelying 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
$$D = F\langle a,b,c,d\rangle / \langle f(a),g(b),h(c),u(d),a+b+c+d\rangle$$ has quadratic growth (in fact, $$2n+1$$ 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.
 * Let $$F$$ be a field and let $$f,g,h,u\in F[x]$$ be irreducible polynomials of degree 2. The algebra:

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 $$A$$?