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V. V. Bavula

Publications and source records attributed to V. V. Bavula.

At least 19 recordsLinked to original sources
arXiv2026

On minimal noncommutative rings

We study minimal noncommutative rings, that is noncommutative rings whose proper subrings and homomorphic images are all commutative. These rings were introduced by Bell and Danchev in order to test commutativity theorems. They raised the problems of describing all such rings in the finite and infinite cases. In the finite case, we give a classification into three pairwise disjoint classes, the first two of which are completely characterised. For the third class, we give a finite procedure which can produce any of (and only) the required rings. We also translate the problem into commutative algebra, in terms of finite local rings with small socle and a kind of cancellation property. Finally, we show that if an infinite minimal noncommutative ring exists, then it is a division algebra with very strange properties, and a counterexample to several longstanding conjectures.

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arXiv2026

Explicit descriptions of the subfields $(NL)^{pi}$ and $(NL)^{pi}(NL)^{sep}$ of $NL$ and new explicit criteria for $NL = (NL)^{pi}(NL)^{sep}$

Let $L=K(θ)\simeq K[x]/f(x)$ be a simple field extension in prime characteristic $p>0$, $L^{sep}$ and $L^{pi}$ be the maximal separable and purely inseparable subfields of $L$, respectively. Let $N/K$ be a purely inseparable field extension. For the field extensions $L/K$ and $NL/N$, the aim of the paper is to give explicit descriptions of the following subfields and their degrees in terms of the coefficients of the polynomial $f$ and two numerical field invariants $m_f$ and $m_{f,N}$: $L^{pi}$, $L^{pi}L^{sep}$, $(NL)^{pi}$ and $(NL)^{pi}(NL)^{sep}$. From these results, we derive new explicit criteria for $L=L^{pi}L^{sep}$ and $NL=(NL)^{pi}(NL)^{sep}$.

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arXiv2026

Analogue of the Galois Theory for normal fields and B-extensions (characteristic free approach)

The aim of the paper is to introduce B-extensions which are the most symmetrical finite field extensions (a finite field extension $L/K$ is called a {\it B-extension} if the endomorphism algebra ${\rm End}_K(L)$ is generated by the algebra of differential operators ${\cal D} (L/K)$ on the $K$-algebra $L$ and the automorphism group $G(L/K):={\rm Aut}_{K-{\rm alg}}(L)$) and to obtain an analogue of the Galois Theory for B-extensions. Surprisingly, the class of B-extensions coincides with the class of {\it normal } finite field extensions. As a result, an analogue of the Galois Theory is obtained for normal field extensions. In particular, all Galois field extensions and all purely inseparable field extensions are B-extensions. Our approach is a ring theoretic (characteristic free) approach which is based on central simple algebras. In this approach, analogues of the Galois Correspondences (for subfields and normal subfields of $L$) are deduced from the Double Centralizer Theorem which is applied to the central simple algebra ${\rm End}_K(L)$ and subfields of B-extensions. Since Galois finite field extensions are B-extensions, this approach gives a new conceptual (short) proofs of key results of the Galois Theory, see [2] for details. It also reveals that the `maximal symmetry' (of field extensions) is the essence of the classical Galois Theory and the analogue of the Galois Theory for normal field extensions.

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arXiv2026

Analogue of the Galois Theory for arbitrary finite field extensions

This paper is a finishing touch to the (over 200 years) {\em classical} `Galois Theory' of {\em arbitrary} finite field extensions, i.e. the goal of it is to describe intermediate subfields of an arbitrary finite field extension via {\em invariants} of `natural/obvious' objects that are associated with subfields via two Galois-type correspondences. The classical Galois Theory covers the case of finite Galois field extensions. For finite Galois field extensions the objects are their Galois groups and their invariants. In \cite{GaloisTh-RingThAp}, we introduce a new (ring theoretic) approach to the Galois Theory which is based on the {\em principle of maximal symmetry}. In \cite{AnGaloisTh-NORMAL-Fields}, the maximal symmetry of {\em normal} finite field extensions yields an analogue of the Galois Theory for them. For a normal finite field extension $L/K$ the `natural/obvious' objects are the subalgebra $\CD (L/K)\rtimes G(L/K)$ of $\End (L/K)$ that is generated by the automorphism group $G(L/K)$ and the algebra $\CD (L/K)$ of differential operators on $L/K$ and its `invariants'. The `maximal symmetry' means the equality $\End (L/K)=\CD (L/K)\rtimes G(L/K)$ which turns out to be a characteristic property of {\em normal} finite field extensions, \cite{AnGaloisTh-NORMAL-Fields}. The aim of this paper is to obtain an analogue of the Galois Theory for {\em arbitrary} finite field extensions based on results and ideas of \cite{GaloisTh-RingThAp} and \cite{AnGaloisTh-NORMAL-Fields}.

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arXiv2025

The Galois Theory (a ring theoretic approach)

The fundamental concepts in the Galois Theory are separable, normal and Galois field extensions. These concepts are central in proofs of the Galois Theory. In the paper, we introduce a new approach, a ring theoretic approach, to the Galois Theory which is based on central simple algebras and none of the above concepts are used or even mentioned. The only concept which is used is `G-extension' (a finite field extension $L/K$ is called a {\em G-extension} if the endomorphism algebra ${\rm End}_K(L)$ is generated by the field $L$ and the automorphism group ${\rm Aut}_{K-{\rm alg}}(L)$). So, G-extensions are the most symmetric field extensions. In this approach, the Galois Correspondences (for subfields and Galois subfields of $L$) are deduced from the Double Centralizer Theorem which is applied to the central simple algebra ${\rm End}_K(L)$ and G-extensions. Since the class of G-extensions {\em coincides} with the class of Galois extensions, all main results of the Galois Theory are obtained from the `corresponding' results for G-extensions. This approach gives a new conceptual (short) proofs of key results of the Galois Theory. It also reveals that the `maximal symmetry' (of field extensions) is the essence of the Galois Theory.

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arXiv2024

Affirmative answer to the Question of Leroy and Matczuk on injectivity of endomorphisms of semiprime left Noetherian rings with large images

The class of semiprime left Goldie rings is a huge class of rings that contains many large subclasses of rings -- semiprime left Noetherian rings, semiprime rings with Krull dimension, rings of differential operators on affine algebraic varieties and universal enveloping algebras of finite dimensional Lie algebras to name a few. In the paper, `Ring endomorphisms with large images,' {\em Glasg. Math. J.} {\bf 55} (2013), no. 2, 381--390, A. Leroy and J. Matczuk posed the following question: {\em If a ring endomorphism of a semiprime left Noetherian ring has a large image, must it be injective?} The aim of the paper is to give an affirmative answer to the Question of Leroy and Matczuk and to prove the following more general results. {\bf Theorem. (Dichotomy)} {\em Each endomorphism of a semiprime left Goldie ring with large image is either a monomorphism or otherwise its kernel contains a regular element of the ring ($\Leftrightarrow$ its kernel is an essential left ideal of the ring). In general, both cases are non-empty.} {\bf Theorem. } {\em Every endomorphism with large image of a semiprime ring with Krull dimension is a monomorphism.} {\bf Theorem. (Positive answer to the Question of Leroy and Matczuk)} {\em Every endomorphism with large image of a semiprime left Noetherian ring is a monomorphism.}

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arXiv2024

$\D$-locally nilpotent algebras, their ideal structure and simplicity criteria

The class of $\D$-locally nilpotent algebras (introduced in the paper) is a wide generalization of the algebras of differential operators on commutative algebras. Examples includes all the rings $\CD (A)$ of differential operators on commutative algebras (in arbitrary characteristic), all subalgebras of $\CD (A)$ that contain the algebra $A$, the universal enveloping algebras of nilpotent, solvable and semi-simple Lie algebras, the Poisson universal enveloping algebra of an arbitrary Poisson algebra, iterated Ore extensions $A[x_1, \ldots , x_n ; \d_1 , \ldots , \d_n]$, certain generalized Weyl algebras, and others. In \cite{SimCrit-difop}, simplicity criteria are given for the algebras differential operators on commutative algebras (it was a long standing problem). The aim of the paper is to describe the ideal structure of $\D$-locally nilpotent algebras and as a corollary to give simplicity criteria for them (it is a generalization of the results of \cite{SimCrit-difop}). Examples are considered.

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arXiv2024

Ore sets, denominator sets and the left regular left quotient ring of a ring

The aim of the papers is to describe the left regular left quotient ring ${}'Q(R)$ and the right regular right quotient ring $Q'(R)$ for the following algebras $R$: $\mS_n=\mS_1^{\t n}$ is the algebra of one-sided inverses, where $\mS_1=K\langle x,y\, | \, yx=1\rangle$, $\CI_n=K\langle \der_1, \ldots, \der_n,\int_1,\ldots, \int_n\rangle$ is the algebra of scalar integro-differential operators and the Jacobian algebra $\mA_1=K\langle x,\der, (\der x)^{-1}\rangle$. The sets of left and right regular elements of the algebras $\mS_1$, $\CI_1$, $\mA_1$ and $\mI_1=K\langle x, \der,\int\rangle$. A progress is made on the following conjecture, \cite{Clas-lreg-quot}: $${}'Q(\mI_n)\simeq Q(A_n)\;\; {\rm where}\;\; \mI_n =K\bigg\langle x_1,\ldots , x_n, \der_1, \ldots, \der_n,\int_1,\ldots, \int_n\bigg\rangle$$ is the algebra of polynomial integro-differential operators and $Q(A_n)$ is the classical quotient ring (of fractions) of the $n$'th Weyl algebra $A_n$, i.e. a criterion is given when the isomorphism holds. We produce several general constructions of left Ore and left denominator sets that appear naturally in applications and are of independent interest and use them to produce explicit left denominator sets that give the localization ring isomorphic to ${}'Q(\mS_n)$ or ${}'Q(\mI_n)$ or ${}'Q(\mA_n)$ where $\mA_n:=\mA_1^{\t n}$. Several characterizations of one-sided regular elements of a ring are given in module-theoretic and one-sided-ideal-theoretic way.

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arXiv2022

The Question of Arnold on classification of co-artin subalgebras in singularity theory

In \cite[Section 5, p.32]{Arnold-1998}, Arnold writes: "Classification of singularities of curves can be interpreted in dual terms as a description of 'co-artin' subalgebras of finite co-dimension in the algebra of formal series in a single variable (up to isomorphism of the algebra of formal series)." In the paper, such a description is obtained but up to isomorphism of algebraic curves (i.e. this description is finer). Let $K$ be an algebraically closed field of arbitrary characteristic. The aim of the paper is to give a classification (up to isomorphism) of the set of subalgebras $\mathcal{A}$ of the polynomial algebra $K[x]$ that contains the ideal $x^mK[x]$ for some $m\geq 1$. It is proven that the set $\mathcal{A} = \coprod_{m, Γ}\mathcal{A} (m, Γ)$ is a disjoint union of affine algebraic varieties (where $Γ\coprod \{0, m, m+1, \ldots \}$ is the semigroup of the singularity and $m-1$ is the Frobenius number). It is proven that each set $\mathcal{A} (m, Γ)$ is an affine algebraic variety and explicit generators and defining relations are given for the algebra of regular functions on $\mathcal{A} (m ,Γ)$. An isomorphism criterion is given for the algebras in $\mathcal{A}$. For each algebra $A\in \mathcal{A} (m, Γ)$, explicit sets of generators and defining relations are given and the automorphism group ${\rm Aut}_K(A)$ is explicitly described. The automorphism group of the algebra $A$ is finite iff the algebra $A$ is not isomorphic to a monomial algebra, and in this case $|{\rm Aut}_K(A)|<{\rm dim}_K(A/\mathfrak{c}_A)$ where $\mathfrak{c}_A$ is the conductor of $A$. The set of orders of the automorphism groups of the algebras in $\mathcal{A} (m , Γ)$ is explicitly described.

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arXiv2021

Localizable sets and the localization of a ring at a localizable set

The concepts of localizable set, localization of a ring and a module at a localizable set are introduced and studied. Localizable sets are generalization of Ore sets and denominator sets, and the localization of a ring/module at a localizable set is a generalization of localization of a ring/module at a denominator set. For a semiprime left Goldie ring, it is proven that the set of maximal left localizable sets that contain all regular elements is equal to the set of maximal left denominator sets (and they are explicitly described). For a semiprime Goldie ring, it is proven that the following five sets coincide: the maximal Ore sets, the maximal denominator sets, the maximal left or right or two-sided localizable sets that contain all regular elements (and they are explicitly described).

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arXiv2021

Holonomic modules and 1-generation in the Jacobian Conjecture

A polynomial endomorphism $σ\in {\rm End}_K(P_n)$ is called a Jacobian map if its Jacobian is a nonzero scalar (the field has zero characteristic). Each Jacobian map $σ$ is extended to an endomorphism $σ$ of the Weyl algebra $A_n$. The Jacobian Conjecture (JC) says that every Jacobian map is an automorphism. Clearly, the Jacobian Conjecture is true iff the twisted (by $σ$) $P_n$-module ${}^σ P_n$ is 1-generated for all Jacobian maps $σ$. It is shown that the $A_n$-module ${}^σ P_n$ is 1-generated for all Jacobian maps $σ$. Furthermore, the $A_n$-module ${}^σ P_n$ is holonomic and as a result has finite length. An explicit upper bound is found for the length of the $A_n$-module ${}^σ P_n$ in terms of the degree ${\rm deg} (σ)$ of the Jacobian map $σ$. Analogous results are given for the Conjecture of Dixmier and the Poisson Conjecture. These results show that the Jacobian Conjecture, the Conjecture of Dixmier and the Poisson Conjecture are questions about holonomic modules for the Weyl algebra $A_n$, the images of the Jacobian maps, endomorphisms of the Weyl algebra $A_n$ and the Poisson endomorphisms are large in the sense that further strengthening of the results on largeness would be either to prove the conjectures or produce counter examples. A short direct algebraic (without reduction to prime characteristic) proof is given of equivalence of the Jacobian and the Poisson Conjectures (this gives a new short proof of equivalence of the Jacobian, Poisson and Dixmier Conjectures).

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arXiv2021

Isomorphism problems and groups of automorphisms for Ore extensions $K[x][y; f\frac{d}{dx} ]$ (prime characteristic)

Let $Λ(f) = K[x][y; f\frac{d}{dx} ]$ be an Ore extension of a polynomial algebra $K[x]$ over an arbitrary field $K$ of characteristic $p>0$ where $f\in K[x]$. For each polynomial $f$, the automorphism group of the algebras $Λ(f)$ is explicitly described. The automorphism group ${\rm Aut}_K(Ł(f))=S\rtimes G_f$ is a semidirect product of two explicit groups where $G_f$ is the {\em eigengroup} of the polynomial $f$ (the set of all automorphisms of $K[x]$ such that $f$ is their common eigenvector). For each polynomial $f$, the eigengroup $G_f$ is explicitly described. It is proven that every subgroup of ${\rm Aut}_K(K[x])$ is the eigengroup of a polynomial. It is proven that the Krull and global dimensions of the algebra $Λ(f)$ are 2. The prime, completely prime, primitive and maximal ideals of the algebra $Λ(f)$ are classified.

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arXiv2021

Isomorphism problems and groups of automorphisms for Ore extensions $K[x][y; δ]$ (zero characteristic)

Let $Ł(f) = K[x][y; f\frac{d}{dx} ]$ be an Ore extension of a polynomial algebra $K[x]$ over a field $K$ of characteristic zero where $f\in K[x]$. For a given polynomial $f$, the automorphism group of the algebra $Ł(f) $ is explicitly described. The polynomial case $Ł(0) = K[x,y]$ and the case of the Weyl algebra $A_1= K[x][y; \frac{d}{dx} ]$ were done done by Jung (1942) and van der Kulk (1953), and Dixmier (1968), respectively. In 1997, Alev and Dumas proved that the algebras $Ł(f)$ and $Ł(g)$ are isomorphic iff $g(x) = łf(αx+β)$ for some $ł, α\in K\backslash \{ 0\}$ and $β\in K$. In 2015, Benkart, Lopes and Ondrus gave a complete description of the set of automorphism groups of algebras $Ł(f)$. In this paper we complete the picture, i.e. {\em given} the polynomial $f$ we have the explicit description of the automorphism group of $Ł(f)$.

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arXiv2021

The Poisson enveloping algebra and the algebra of Poisson differential operators of a generalized Weyl Poisson algebra

For a generalized Weyl Poisson algebra $A$, explicit sets of generators and defining relations are presented for its Poisson enveloping algebra $\CU (A)$. Simplicity criteria are given for the algebra $\CU (A)$ and algebra of Poisson differential operators $P\CD (A)$ on $A$. The Gelfand-Kirillov dimensions of the algebras $\CU (A)$ and $P\CD (A)$ are calculated. It is proven that the algebra $\CU (A)$ is a domain provided that the coefficient ring $D$ of the generalized Weyl Poisson algebra $A$ is a domain of essentially finite type over a perfect field. For the algebra $A$, the set of its minimal primes and the prime radical are described and an equidimensionality criterion is given. For the equidimensional algebra $A$ of essentially finite type, two regularity criteria are presented.

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arXiv2021

The PBW Theorem and simplicity criteria for the Poisson enveloping algebra and the algebra of Poisson differential operators

For an arbitrary Poisson algebra $\CP$ over an arbitrary field, an (analogue of) the Poincaré-Birkhof-Witt Theorem is proven and several presentations/constructions for its Poisson enveloping algebra $\CU (\CP )$ are given. As a result, explicit sets of generators and defining relations are given for $\CU (\CP )$ and the algebra $P\CD (\CP)$ of Poisson differential operators on $\CP$. Simplicity criteria for the algebras $\CU (\CP )$ and $P\CD (\CP )$ are given. In the case when the algebra $\CP$ is of essentially finite type, a criterion for the algebra $\CU (\CP )$ to be a domain is presented and a criterion for a natural epimorphism $\CU (\CP )\ra P\CD (\CP )$ to be an isomorphism is given. The kernel of the epimorphism is described and for large classes of Poisson algebras an explicit set of generators is given. Explicit formulae for the Gelfand-Kirillov dimension of the algebras $\CU (\CP )$ and $P\CD (\CP)$ are given. In the case when the Poisson algebra $\CP$ is a regular domain of essentially finite type an explicit simplecticity criterion for $\CP$ is found and a criterion is presented for the algebra $\CU (\CP )$ to be isomorphic to the algebra $\CD (\CP )$ of differential operators on $\CP$.

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arXiv2021

Generators and defining relations for the ring of differential operators on a smooth affine algebraic variety

For the ring of differential operators on a smooth affine algebraic variety $X$ over a field of characteristic zero a finite set of algebra generators and a finite set of defining relations are found explicitly. As a consequence, a finite set of generators and a finite set of defining relations are given for the module $\Der_K(\OO (X))$ of derivations on the algebra $\OO (X)$ of regular functions on the variety $X$. For the variety $X$ which is not necessarily smooth, a set of natural derivations ${\rm der}_K(\OO (X))$ of the algebra $\OO (X)$ and a ring $\gD (\OO (X))$ of natural differential operators on $\OO (X)$ are introduced. The algebra $\gD (\OO (X))$ is a Noetherian algebra of Gelfand-Kirillov dimension $2\dim (X)$. When $X$ is smooth then ${\rm der}_K(\OO (X))=\Der_K(\OO (X))$ and $\gD (\OO (X))=\CD (\OO (X))$. A criterion of smoothness of $X$ is given when $X$ is irreducible ($X$ is smooth iff $\gD (\OO (X))$ is a simple algebra iff $\OO (X)$ is a simple $\gD (\OO (X))$-module). The same results are true for regular algebras of essentially finite type. For a singular irreducible affine algebraic variety $X$, in general, the algebra of differential operators $\CD (\OO (X))$ needs not be finitely generated nor (left or right) Noetherian, it is proved that each term $\CD (\OO (X))_i$ of the order filtration $\CD (\OO (X))=\cup_{i\geq 0}\CD (\OO (X))_i$ is a finitely generated left $\OO (X)$-module.

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arXiv2019

Simplicity criteria for rings of differential operators

Let $K$ be a field of arbitrary characteristic, $\CA$ be a commutative $K$-algebra which is a domain of essentially finite type (eg, the algebra of functions on an irreducible affine algebraic variety), $\ga_r$ be its {\em Jacobian ideal}, $\CD (\CA )$ be the algebra of differential operators on the algebra $\CA$. The aim of the paper is to give a simplicity criterion for the algebra $\CD (\CA )$: {\em The algebra $\CD (\CA )$ is simple iff $\CD (\CA ) \ga_r^i\CD (\CA )= \CD (\CA )$ for all $i\geq 1$ provided the field $K$ is a perfect field.} Furthermore, a simplicity criterion is given for the algebra $\CD (R)$ of differential operators on an arbitrary commutative algebra $R$ over an arbitrary field. This gives an answer to an old question to find a simplicity criterion for algebras of differential operators.

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