Stable nil algebras of finite Gelfand–Kirillov dimension and the completion of stability-preserving embedding theorems

Alahmadi, Alsulami, Jain and Zelmanov (Trans. Amer. Math. Soc. 372, 2019) introduced matrix wreath products of algebras and used them to prove embedding theorems for Jacobson radical, nil, and primitive algebras, and to construct finitely generated nil algebras of every Gelfand–Kirillov (GK) dimension at least 8 over countable fields. The paper itself records what its program still lacks. First, it is unknown whether there exists an infinite-dimensional, finitely generated, nil algebra of finite GK dimension that is stable nil, i.e., all of whose matrix algebras $M_n(A)$ are nil — a finitistic instance of Köthe's problem. Such an example would immediately yield the paper's conditional strengthenings: embeddings of every countable-dimensional stable nil algebra of finite GK dimension into a finitely generated stable nil algebra of finite GK dimension, and embeddings of every countable-dimensional stable algebraic primitive algebra of finite GK dimension into a 2-generated stable algebraic primitive algebra of finite GK dimension, fully answering Question 7(ii) of Bell–Small–Smoktunowicz. Second, all growth-controlled embedding theorems of the paper assume a countable ground field, and no infinite-dimensional finitely generated nil (or algebraic) algebra of polynomially bounded growth is known over an uncountable field. Both questions remain open in the literature. Later work (Greenfeld–Zelmanov 2023; Greenfeld 2025) extends the growth landscape of nil algebras — improving the realizable GK-dimension spectrum from $[8,\infty)$ to $[6,\infty)$ over countable fields — but never addresses stability under matrix extensions or the countability barrier.

Background

A classical theme in algebra asks when a countable-dimensional structure can be embedded into a finitely generated one while preserving its structural properties. For associative algebras over a field it has been known since the 1950s that every countable-dimensional algebra embeds into a finitely generated one (an analogue of the classical group-theoretic embedding theorem), but preserving properties such as being Jacobson radical (i.e., coinciding with its own Jacobson radical, the intersection of the annihilators of all irreducible modules), nil, or primitive, while simultaneously controlling growth, is far harder. To address this, Alahmadi, Alsulami, Jain and Zelmanov (2019) introduced the matrix wreath product of algebras — an analogue of the wreath product of groups, built from an algebra $B$ and the space of linear transformations from the unital hull of $B$ to its tensor product with $A$ — and used it to prove embedding theorems for radical, nil and primitive algebras.

The basic definitions are as follows. An associative algebra is nil if every element is nilpotent, and algebraic if every element satisfies a nonzero polynomial over the ground field. The Kurosh problem asks whether every finitely generated algebraic algebra is finite-dimensional; Golod's classical counterexamples (obtained from Golod–Shafarevich presentations with infinitely many tailored relations) are finitely generated, infinite-dimensional nil algebras, but of exponential growth. Growth is measured by the Gelfand–Kirillov (GK) dimension: for a finitely generated algebra $A$ with finite-dimensional generating subspace $V$, the GK dimension is $\limsup_n \log \dim(V^n)/\log n$, so finite GK dimension means polynomially bounded growth. Whether a finitely generated, infinite-dimensional nil algebra of finite GK dimension exists was a question of Small, answered by Lenagan and Smoktunowicz (2007) over any countable field (GK dimension at most 20, refined to at most 3, with two generators, by Lenagan–Smoktunowicz–Young 2012). All known examples are built over countable fields, and the LSY construction, in its authors' words, depends crucially on countability; over uncountable fields the best known infinite-dimensional finitely generated nil algebras (Bell–Young 2011) still have super-polynomial growth.

A nil algebra $A$ is called stable nil if every matrix algebra $M_n(A)$, $n \geq 1$, is nil (similarly, stable algebraic if every $M_n(A)$ is algebraic). By classical work of Köthe (1930), in the equivalent formulations organized by Krempa (1972), the assertion that every nil algebra is stable nil is equivalent to Köthe's problem — whether the sum of two nil left ideals of an associative ring is always nil — one of the famous open problems of ring theory. Nilpotent algebras and locally nilpotent algebras are trivially stable nil; however, a finitely generated nilpotent algebra is finite-dimensional, and a finitely generated locally nilpotent algebra is nilpotent, so any infinite-dimensional finitely generated stable nil algebra must be genuinely non-locally-nilpotent. The source paper constructs a finitely generated stable nil algebra by a Golod–Shafarevich argument tensored with an infinite-variable polynomial algebra, but this construction gives no finite GK-dimension bound, and the growth control in its embedding theorems comes from using the Lenagan–Smoktunowicz–Young algebra, which is not known to be stable nil.

An algebra is primitive if it has a faithful irreducible module. A question of Kaplansky, recalled in Bell–Small–Smoktunowicz (2012), asks whether an infinite-dimensional, finitely generated, algebraic primitive algebra exists; examples were first constructed in the early 2000s, and Bell–Small–Smoktunowicz constructed, over any countable field, finitely generated algebraic primitive algebras of finite GK dimension and, separately, a two-generated algebraic primitive algebra (their Theorem 1.2). Their Question 7 asks for the two-generated construction over an uncountable field (part (i)) and with finite GK dimension (part (ii)). Here $M_\infty(A)$ denotes the algebra of countably-indexed matrices over $A$ with finitely many nonzero entries; an algebra is $M_\infty$-embeddable in $B$ (as a left ideal) if there is an embedding $M_\infty(A) \to B$ whose image is a left ideal.

Within this framework the source paper proves: every countable-dimensional Jacobson radical algebra, and every countable-dimensional stable nil algebra, embeds into a finitely generated algebra of the same kind (Theorems 4.1, 4.3); with GK-dimension control (at most $d+6$, over a countable ground field) the target is only guaranteed nil, not stable nil; every countable-dimensional stable algebraic primitive algebra is $M_\infty$-embeddable as a left ideal in a 2-generated algebraic primitive algebra (Theorem 5.1, answering Question 7(i)); and, with GK-dimension control, in a finitely generated algebraic primitive algebra of GK dimension at most $d+6$ (Theorem 5.2, over a countable field). The paper records precisely what is missing: two remarks state that a stable nil Lenagan–Smoktunowicz algebra would yield the stable, growth-controlled versions of these embedding theorems — including the 2-generated stable algebraic primitive targets answering Question 7(ii) — and all growth-controlled statements assume a countable ground field. Later work (Greenfeld–Zelmanov 2023; Greenfeld 2025) extends the wreath-product machinery to realize prescribed and oscillating growth rates of nil algebras and widens the realizable GK-dimension spectrum, but does not address stability under matrix extensions; as of 2025 it is still unknown whether any infinite-dimensional finitely generated algebraic algebra of polynomially bounded growth exists over an uncountable field.

Problem Statement

The problem, as recorded by Alahmadi–Alsulami–Jain–Zelmanov, has one common scientific objective — extending their embedding theorems to their full form: stability-preserving, growth-controlled, and valid over arbitrary ground fields — with two independent, explicitly identified obstructions.

(1) Primary question (stability). Does there exist an infinite-dimensional, finitely generated (affine) nil algebra $A$ of finite Gelfand–Kirillov dimension, over some field (for the embedding applications, over any prescribed countable field), that is stable nil, i.e., such that every matrix algebra $M_n(A)$, $n \geq 1$, is nil?

(2) Linked consequences to derive (the source paper's conditional strengthenings, obtained from its matrix wreath product machinery once a suitable auxiliary algebra exists). Given a positive answer to (1), derive: (a) every countable-dimensional stable nil algebra of finite GK dimension over a countable field embeds into a finitely generated stable nil algebra of finite GK dimension; (b) every countable-dimensional stable algebraic primitive algebra of finite GK dimension over a countable field is $M_\infty$-embeddable as a left ideal in a 2-generated stable algebraic primitive algebra of finite GK dimension, thereby fully answering Question 7(ii) of Bell–Small–Smoktunowicz.

(3) Second component (field generality). The growth-controlled embedding theorems of the source paper (Theorems 4.2, 4.3, 5.2, 6.1) all assume a countable ground field. Does there exist, over some uncountable field, an infinite-dimensional finitely generated nil algebra of finite GK dimension — more generally, an infinite-dimensional finitely generated algebraic algebra of polynomially bounded growth — and, if so, do the embedding theorems extend to arbitrary ground fields? Or is the countability hypothesis essential?

A positive solution of Köthe's problem would immediately resolve (1) affirmatively (every nil algebra, in particular a Lenagan–Smoktunowicz algebra, would then be stable nil), so question (1) is a finitistic instance of Köthe's problem, strictly weaker than the full conjecture. Components (1)–(2) and (3) are independent of each other, but they jointly constitute the open content the source paper records for its embedding program.

References

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