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V. A. Bovdi

Publications and source records attributed to V. A. Bovdi.

At least 19 recordsLinked to original sources
arXiv2024

Rota--Baxter and averaging operators on racks and rack algebras

In the present article we define and investigate relative Rota--Baxter operators and relative averaging operators on racks and rack algebras. Also, if B is a Rota--Baxter or averaging operator on a rack X, then we can extend B by linearity to the rack algebra k[X]. On the other side, we have definitions of Rota--Baxter and averaging operators on arbitrary algebra. We find connections between these operators. In particular, we prove that if B : X --> X is an averaging operator on a rack, then its linear extension on a rack algebra k[X] gives an averaging operator.

math.RA↗
arXiv2022

Orbits of actions of group superschemes

Working over an algebraically closed field $\Bbbk$, we prove that all orbits of a left action of an algebraic group superscheme $G$ on a superscheme $X$ of finite type are locally closed. Moreover, such an orbit $Gx$, where $x$ is a $\Bbbk$-point of $X$, is closed if and only if $G_{ev}x$ is closed in $X_{ev}$, or equivalently, if and only if $G_{res}x$ is closed in $X_{res}$. Here $G_{ev}$ is the largest purely even group super-subscheme of $G$ and $G_{res}$ is $G_{ev}$ regarded as a group scheme. Similarly, $X_{ev}$ is the largest purely even super-subscheme of $X$ and $X_{res}$ is $X_{ev}$ regarded as a scheme. We also prove that $\mathrm{sdim}(Gx)=\mathrm{sdim}(G)-\mathrm{sdim}(G_x)$, where $G_x$ is the stabilizer of $x$.

math.RT↗
arXiv2021

Generating solutions of a linear equation and structure of elements of the Zelisko group

Solutions of a linear equation b=ax in a homomorphic image of a commutative Bezout domain of stable range 1.5 is developed. It is proved that the set of solutions of a solvable linear equation contains at least one solution that divides the rest, which is called a generating solution. Generating solutions are pairwise associates. Using this result, the structure of elements of the Zelisko group is investigated.

math.RA↗
arXiv2021

Modules over some group rings having d-generator property

For modules over group rings we introduce the following numerical parameter. We say that a module A over a ring R has finite r-generator property if each f.g. (finitely generated) R-submodule of A can be generated exactly by r elements and there exists a f.g. R-submodule D of A, which has a minimal generating subset, consisting exactly of r elements. Let FG be the group algebra of a finite group G over a field F. In the present paper modules over the algebra FG having finite generator property are described.

math.AC↗
arXiv2021

Some ranks of modules over group rings

A commutative ring R has finite rank r, if each ideal of R is generated at most by r elements. A commutative ring R has the r-generator property, if each finitely generated ideal of R can be generated by r elements. Such rings are closely related to Prüfer domains. In the present paper we investigate some analogs of these concepts for modules over group rings.

math.AC↗
arXiv2020

Subgroups of a finitary linear group

Let FL_s(K) be the finitary linear group of degree s over an associative ring K with unity. We prove that the torsion subgroups of FL_s(K) are locally finite for certain classes of rings K. A description of some f.g. solvable subgroups of FL_s(K) are given.

math.GR↗
arXiv2017

Completely simple endomorphism rings of modules

It is proved that if A_p is a countable elementary abelian p-group, then: (i) The ring End(A_p) does not admit a nondiscrete locally compact ring topology. (ii) Under (CH) the simple ring End(A_p)/I, where I is the ideal of End(A_p) consisting of all endomorphisms with finite images, does not admit a nondiscrete locally compact ring topology. (iii) The finite topology on End(A_p) is the only second metrizable ring topology on it. Moreover, a characterization of completely simple endomorphism rings of the endomorphism rings of modules over commutative rings is also obtained.

math.RA↗
arXiv2017

Reduction of a pair of skew-symmetric matrices to its canonical form under congruence

Let $(A,B)$ be a pair of skew-symmetric matrices over a field of characteristic not 2. Its regularization decomposition is a direct sum \[ (\underline{\underline A},\underline{\underline B})\oplus (A_1,B_1)\oplus\dots\oplus(A_t,B_t) \] that is congruent to $(A,B)$, in which $(\underline{\underline A},\underline{\underline B})$ is a pair of nonsingular matrices and $(A_1,B_1),$ $\dots,$ $(A_t,B_t)$ are singular indecomposable canonical pairs of skew-symmetric matrices under congruence. We give an algorithm that constructs a regularization decomposition. We also give a constructive proof of the known canonical form of $(A,B)$ under congruence over an algebraically closed field of characteristic not 2.

math.RT↗
arXiv2017

Unitary and symmetric units of a commutative group algebra

Let F be the field of two elements and G a finite abelian 2-group with an involutory automorphism. The extension of this automorphism to the group algebra FG is called an involutory involution. This determines the groups of unitary and symmetric normalized units of FG. We calculate the orders and the invariants of these subgroups.

math.RA↗
arXiv2010

Torsion units in integral group rings of Janko simple groups

Using the Luthar--Passi method, we investigate the classical Zassenhaus conjecture for the normalized unit group of integral group rings of Janko sporadic simple groups. As a consequence, we obtain that the Gruenberg-Kegel graph of the Janko groups $J_1$, $J_2$ and $J_3$ is the same as that of the normalized unit group of their respective integral group ring.

math.GR↗
arXiv2009

Group algebras whose group of units is powerful

A p-group is called powerful if every commutator is a product of pth powers when p is odd and a product of fourth powers when p=2. In the group algebra of a group G of p-power order over a finite field of characteristic p, the group of normalized units is always a p-group. We prove that it is never powerful except, of course, when G is abelian.

math.RA↗
arXiv2008

Integral group ring of Rudvalis simple group

Using the Luthar-Passi method, we investigate the classical Zassenhaus conjecture for the normalized unit group of the integral group ring of the Rudvalis sporadic simple group Ru. As a consequence, for this group we confirm Kimmerle's conjecture on prime graphs.

math.RA↗
arXiv2008

Normality in group rings

Let $KG$ be the group ring of a group $G$ over a commutative ring $K$ with unity. The rings $KG$ are described for which $xx^σ=x^σx$ for all $x=\sum_{g\in G}α_gg\in KG$, where \quad $x\mapsto x^σ=~\sum_{g\in G}α_gf(g)σ(g)$\quad is an involution of $KG$; here $f: G\to U(K)$ is a homomorphism and $σ$ is an anti-automorphism of order two of $G$.

math.RA↗
arXiv2008

Integral group ring of the Suzuki sporadic simple group

Using the Luthar--Passi method, we investigate the classical Zassenhaus conjecture for the normalized unit group of the integral group ring of the Suzuki sporadic simple group Suz. As a consequence, for this group we confirm the Kimmerle's conjecture on prime graphs.

math.RA↗
arXiv2007

Symmetric units in modular group algebras

Let p be a prime, G a locally finite p-group, K a commutative ring of characteristic p. The anti-automorphism g\mapsto g\m1 of G extends linearly to an anti-automorphism a\mapsto a^* of KG. An element a of KG is called symmetric if a^*=a. In this paper we answer the question: for which G and K do the symmetric units of KG form a multiplicative group.

math.RA↗