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In the present paper, necessary and sufficient conditions are given for the equality of the power rezidue symbols $$ {\left( {\frac{\alpha }{a}} \right)_n} $$ and $$ {\left( {\frac{\alpha }{a}} \right)_n} $$ in the cyclotomic field ℚ(ζ n), 2 ∤ n, for a ∈ ℤ, ( a, n) = 1. This result is a generalization of the classical Eisenstein reciprocity law and its continuation in a Hasse’s paper. Bibliography: 3 titles. [ABSTRACT FROM AUTHOR]

In this paper, an analog of the conformable fractional derivative is defined in an arbitrary finite-dimensional commutative associative algebra. Functions taking values in the indicated algebras and having derivatives in the sense of a conformable fractional derivative are called φ-monogenic. A relation between the concepts of φ-monogenic and monogenic functions in such algebras has been established. Two new definitions have been proposed for the fractional derivative of the functions with values in finite-dimensional commutative associative algebras. [ABSTRACT FROM AUTHOR]

Let 풜 be an algebra over a field 픽 generated by a set of matrices 풮. The paper considers algorithmic aspects of checking whether 풜 coincides with the full matrix algebra. Laffey has shown that for 픽 = ℂ, under the assumption that 풮 contains a Jordan matrix from a certain class, there is a fast method for checking whether 풜 possesses nontrivial invariant subspaces and, consequently, coincides with the full algebra by Burnside's theorem. This paper extends the class to the largest subclass of Jordan matrices on which the algorithm works correctly. Examples demonstrating different types of behavior of other matrix systems are provided. [ABSTRACT FROM AUTHOR]

This paper presents variants of the topological Jacobson radical of algebras in quasi-regular and modular versions of the definition, which use the ideas of the construction of the Brown–McCoy radical and the descriptions of the Jacobson radical of alternative algebras. [ABSTRACT FROM AUTHOR]

In this survey paper, we collect properties of one-relator algebras in varieties of linear algebras. The main attention is paid to Schreier varieties of algebras. A variety of linear algebras is said to be Schreier if any subalgebra of a free algebra of this variety is free. [ABSTRACT FROM AUTHOR]

In this paper, numerical characteristics of identities with involution of nonassociative algebras are studied. We construct a series of examples of algebras with overexponential growth of codimensions with involution. [ABSTRACT FROM AUTHOR]

In this paper, we consider a metric graph of the simplest structure, which is a three-ray star with edges of pairwise different lengths. For it, all steps of the procedure which reduces the algebra of the eikonals of the graph to a canonical form, are described in detail. Using this form, the spectrum of the eikonal algebra is endowed with a graph structure. [ABSTRACT FROM AUTHOR]

In this paper, we discuss an important and nontrivial theorem on evaluation homomorphisms. We state this theorem as a canonical duality between the family of all smooth mappings f ∈ Hom(M,M′) of a smooth real finite-dimensional manifold M into a similar manifold M′ and the family of homomorphisms φ of the algebra C∞ (M′) of smooth scalar-valued functions on M′ into the analogous algebra C∞ (M) on M, φ ∈ Hom (C∞ (M′), C∞ (M)). This formulation possesses the maximum natural generality and, at the same time, allows it to be used in applications in the standard canonical form. [ABSTRACT FROM AUTHOR]

The paper is devoted to the problem of determining the number of subalgebras in the matrix algebra that are generated by nonderogatory matrices and are distinct up to similarity. A similarity criterion in terms of isomorphism of quotient algebras of the algebra of polynomials for matrix algebras of the type indicated is established. The existence of infinite fields for which the number of distinct algebras is infinite for all values of the matrix order is proved. For algebraically closed fields, the field of reals, and finite fields of sufficiently large cardinalities, the number of distinct algebras generated by nonderogatory matrices is determined as a function of the matrix order. [ABSTRACT FROM AUTHOR]

The paper interrelates the simultaneous triangularization problem for matrix pairs with the Paz problem and known results on the length of the matrix algebra. The length function is applied to the Al'pin–Koreshkov algorithm, and it is demonstrated how its multiplicative complexity can be reduced. An asymptotically superior procedure for verifying the simultaneous triangularizability of a pair of complex matrices is provided. The procedure is based on results on the lengths of upper triangular matrix algebras. Also the definition of the hereditary length of an algebra is introduced, and the problem of computing the hereditary lengths of matrix algebras is discussed. [ABSTRACT FROM AUTHOR]

In this paper, we consider the 핋-space structure of the relatively free Grassmann algebra 픽 (3) without unity over an infinite field of prime and zero characteristic. Our work is focused on 핋-spaces 핎n generated by all so-called n-words. A question about connections between 핎r and 핎n for different natural numbers r and n is investigated. The proved theorem on these connections allows us to construct the diagrams of inclusions, which, to some extent, clarify the structure of the algebra: the basic 핋-spaces produce infinite strictly descending chains of inclusions in the algebra 픽(3). [ABSTRACT FROM AUTHOR]

This paper collects in a unified form well-known versions of the Anderson–Divinsky–Sulinski lemma for algebras that are nearly associative, and gives a number of its analogues for Lie algebras. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity. For positive integers m and n, Anderson and Badawi (Journal of Algebra and Its Applications 16(1):1750013 (21 pp), 2017) defined an ideal I of a ring R to be an (m,n)-closed if whenever x m ∈ I , then x n ∈ I . In this paper we define and study a new generalization of the class of (m,n)-closed ideals which is the class of quasi (m,n)-closed ideals. A proper ideal I is called quasi (m,n)-closed in R if for x ∈ R , x m ∈ I implies either x n ∈ I or x m - n ∈ I . That is, I is quasi (m,n)-closed in R if and only if I is either (m, n)-closed or ( m , m - n )-closed in R. We justify several properties and characterizations of quasi (m,n)-closed ideals with many supporting examples. Furthermore, we investigate quasi (m,n)-closed ideals under various contexts of constructions such as direct products, localizations and homomorphic images. Finally, we discuss the behavior of this class of ideals in idealization rings. [ABSTRACT FROM AUTHOR]

Let A and A ′ be two alternative ∗ -algebras with identities 1 A and 1 A ′ , respectively, and let e 1 and e 2 = 1 A - e 1 be nontrivial symmetric idempotents in A . In this paper, we study the characterization of multiplicative ∗ -Jordan-type maps on alternative algebras. [ABSTRACT FROM AUTHOR]

This paper is devoted to the generalization of the construction of a Jordan algebra of a Lie algebra and the known theorems on the local finite-dimensionality of Lie PI-algebras with an algebraic adjoint representation to Mal'tsev algebras. [ABSTRACT FROM AUTHOR]

The present paper continues a cycle of papers, in which the Yoneda algebras were calculated for several families of algebras of dihedral and semidihedral type in the classification by K. Erdmann. Using the technique of a previous paper, a description of the Yoneda algebras for both families of local algebras occurring in this classification is given. Namely, a conjecture about the structure of the minimal free resolution of a (unique) simple module is stated, which is based on some empirical observations, and after establishing this conjecture, “cohomology information" is derived from the resolution discovered, and, as a result, this allows us to describe the Yoneda algebras of the algebras under consideration, It is noted that a similar technique was applied in computation of the Hochschild cohomology algebra for some finite-dimensional algebras. Bibliography: 23 titles. [ABSTRACT FROM AUTHOR]

This paper starts a series of publications devoted to surveying and developing methods for solving algebraic problems for two-parameter polynomial and rational matrices. The paper considers rank factorizations and, in particular, the relatively irreducible and ΔW-2 factorizations, which are used in solving spectral problems for two-parameter polynomial matrices F(λ, µ). Algorithms for computing these factorizations are suggested and applied to computing points of the regular, singular, and regular-singular spectra and the corresponding spectral vectors of F(λ, µ). The computation of spectrum points reduces to solving algebraic equations in one variable. A new method for computing spectral vectors for given spectrum points is suggested. Algorithms for computing critical points and for constructing a relatively free basis of the right null-space of F(λ, µ) are presented. Conditions sufficient for the existence of a free basis are established, and algorithms for checking them are provided. An algorithm for computing the zero-dimensional solutions of a system of nonlinear algebraic equations in two variables is presented. The spectral properties of the ΔW-2 method are studied. Bibliography: 4 titles. [ABSTRACT FROM AUTHOR]

In this paper, we discuss methods and approaches for definition of multiplication of distributions, which is not defined in general in the classical theory. We show that this problem is related to the fact that the operator of multiplication by a smooth function is nonclosable in the space of distributions. We give the general method of construction of new objects called new distributions, or mnemofunctions, that preserve essential properties of usual distributions and produce algebras as well. We describe various methods of embedding of distribution spaces into algebras of mnemofunctions. All ideas and considerations are illustrated by the simplest example of the distribution space on a circle. Some effects arising in study of equations with distributions as coefficients are demonstrated by example of a linear first-order differential equation. [ABSTRACT FROM AUTHOR]

In this paper, a generalization of composition algebras is given, in which the quadratic form is replaced by a form of arbitrary degree. Two series of generalized composition algebras are found: cyclic algebras of arbitrary order and algebras that are obtained by the generalized doubling procedure applied to cyclic algebras. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new approach to the real realization of spaces over algebras, in which a space over an algebra and the space of its realification have the same dimension. We also give several examples from Euclidean geometry, which illustrate this real realization of spaces over linear algebras. [ABSTRACT FROM AUTHOR]

In this paper we study the class of laterally complete commutative unital regular algebras A over arbitrary fields. We introduce a notion of passport Γ(X) for a faithful regular laterally complete Amodules X, which consist of uniquely defined partition of unity in the Boolean algebra of all idempotents in A and of the set of pairwise different cardinal numbers. We prove that A-modules X and Y are isomorphic if and only if Γ(X) = Γ(Y). Further we study Banach A-modules in the case A = C∞(Q) or A = C∞(Q)+i ·C∞(Q). We establish the equivalence of all norms in a finite-dimensional (respectively, σ-finite-dimensional) A-module and prove an A-version of Riesz Theorem, which gives the criterion of a finite-dimensionality (respectively, σ-finite-dimensionality) of a Banach A-module. [ABSTRACT FROM AUTHOR]

This paper is a sequel of the joint paper by the author with S. O. Ivanov, Yu. Volkov, and G. Zhou. In the present paper, the BV -structure, and therefore, the Gerstenhaber algebra structure on the Hochschild cohomology of local algebras of generalized quaternion type is completely described over a field of characteristic 2. The family of algebras under investigation contains group algebras of generalized quaternion groups for which the case of characteristic 2 is the only one where the calculation of Hochschild cohomology and structures on it is a highly nontrivial problem. Also the group algebras of generalized quaternion groups represent classes of Morita-equivalence of tame group blocks from K. Erdmann's classification. In particular, the BV -structure on the Hochschild cohomology of group algebras of some noncommutative groups is described. [ABSTRACT FROM AUTHOR]

In this paper, we give a general insight into the ideas that make ground for the developing of universal algebraic geometry and logical geometry. We specify the role of algebraic logic as one of the major instruments of the whole theory. The problem of the sameness of geometries of algebraic and definable sets for different algebras is considered as ans illuminating example how algebra, geometry, model theory, and algebraic logic work together. [ABSTRACT FROM AUTHOR]

The paper considers the problem of computing the lengths of matrix incidence algebras over a field whose cardinality is strictly less than the matrix order n. For n = 3, 4, the lengths of all such algebras over the field of two elements are determined. In the case where the ground field and the number n are arbitrary but the Jacobson radical of the algebra has nilpotency index 2, an upper bound for the length is provided. In addition, the incidence algebras isomorphic to a direct sum of triangular matrix algebras of order 2 and an algebra of diagonal matrices are considered. It is shown that the lengths of these algebras over the field of two elements can take only two distinct values, which can be determined exactly. Moreover, the diagonal number of a matrix incidence algebra is introduced and bounded above. [ABSTRACT FROM AUTHOR]

The paper continues the series of papers devoted to surveying and developing methods for solving problems for two-parameter polynomial and rational matrices. Different types of factorizations of two-parameter rational matrices (including irreducible and minimal ones), methods for computing them, and their applications to solving spectral problems are considered. Bibliography: 6 titles. [ABSTRACT FROM AUTHOR]

The paper is a continuation the previous paper, where the lattice of overgroups of the diagonal subgroup in the hyperbolic unitary group for an infinite or sufficiently large simple Artin ring is described. Here the same problem is considered for the hyperbolic unitary group of the second degree over a finite simple Artin ring with a small center. Conditions under which the above lattice admits the standard description are stated. Bibliography: 11 titles. [ABSTRACT FROM AUTHOR]

In this paper, Jordan-Kronecker invariants are calculated for all nilpotent 6- and 7-dimensional Lie algebras. We consider the Poisson bracket family, depending on the lambda parameter on a Lie coalgebra, i.e., on the linear space dual to a Lie algebra. For some space g proposed in the paper, two skew-symmetric matrices are defined for all points x on this linear space. To understand the behaviour of the matrix pencil (A - [lambda][BETA])([chi]), we consider Jordan-Kronecker invariants for this pencil and how they change with [chi] (the latter is done for 6-dimensional Lie algebras)., 1. Basic Definitions and Theorems Definition 1. A Poisson bracket is a bilinear skew-symmetric operation over functions f,g [right arrow] {f,g} satisfying the Jacobi identity and the Leibniz rule. The [...]

This paper is a survey of recent works on invariants of actions of Hopf algebras. Its highlights are results on integrality of H-module PI algebras over subrings of invariant elements obtained by P. Etingof and M. Eryashkin. Older results are also reviewed. [ABSTRACT FROM AUTHOR]

We study the Gnedin–Kingman graph, which corresponds to Pieri's rule for the monomial basis {Mλ} in the algebra QSym of quasisymmetric functions. The paper contains a detailed announcement of results concerning the classification of indecomposable semifinite harmonic functions on the Gnedin–Kingman graph. For these functions, we also establish a multiplicativity property, which is an analog of the Vershik–Kerov ring theorem. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the theorem on the equivalence of the coordinate and algebraic definitions of a smooth manifold. Within the framework of the algebraic approach, a point is considered as a homomorphism from the algebra of smooth real functions defined on a manifold into the field of real numbers. We consider a generalization for the case where the field of real numbers is replaced by an arbitrary associative normalized algebra, generally speaking, noncommutative. [ABSTRACT FROM AUTHOR]

Consider C n as the pseudo-unitary space with the inner product defined by the matrix $$ {p}_n=\left(\begin{array}{l}\kern4em 1\hfill \\ {}\kern3em 1\hfill \\ {}\kern1em \dots \hfill \\ {}1\hfill \end{array}\right) $$ . In this space, centro-unitary matrices play the role of unitary operators. The main result of this paper is a factorization of an arbitrary centro-unitary matrix of even order into a product of simpler centro-unitary matrices. This result is an implication of a similar fact concerning factorizations of pseudo-unitary matrices of the type (n, n). [ABSTRACT FROM AUTHOR]

In this paper, we study an irreducible decomposition structure of the D -module direct image π + (O c n) for the finite map π : C n → C n / (S n 1 × ⋯ × S n r ). We explicitly construct the simple components of π + (O C n) by providing their generators and their multiplicities. Using an equivalence of categories and the higher Specht polynomials, we describe a D -module decomposition of the polynomial ring localized at the discriminant of π . Furthermore, we study the action of invariant differential operators on the higher Specht polynomials. [ABSTRACT FROM AUTHOR]

We introduce a number of definitions of nonfree actions of groups. The most important of them is that of a totally nonfree action; it is naturally related to the theory of characters of groups and their factor representations. This short note is a brief exposition of a part of a more detailed paper on this subject, which is now in preparation. Bibliography: 8 titles. [ABSTRACT FROM AUTHOR]

In this paper, we consider structured automata without output signals whose state sets are endowed with an algebraic structure of hypergraphs. The main result of the paper is a theorem where we obtain necessary and sufficient conditions for the possibility of defining on the state set of some automaton A a structure of a hypergraph H such that the automaton A will be the universal hypergraphic automaton. [ABSTRACT FROM AUTHOR]

The paper introduces the Cayley–Dickson split-sedenion algebra. Exact expressions for the annihilators and orthogonalizers of its zero divisors are obtained, and these results are applied in describing relation graphs of the split-sedenions in terms of their diameters and cliques. [ABSTRACT FROM AUTHOR]

In this paper, we consider various algebras of hypercomplex numbers and geometric structures over them and discuss certain applications of these structures in theoretical physics. [ABSTRACT FROM AUTHOR]

In this paper, we study local and 2-local derivations of p-filiform Leibniz algebras, describe all local derivations of p-filiform non-Lie Leibniz algebras, and also show that there exist two-local derivations on these algebras that are not derivations. [ABSTRACT FROM AUTHOR]

In this paper, we examine topological spaces that can be effectively defined over factor sets modulo equivalences on the set of natural numbers. We formulate a criterion of computable (effective) separability of topological spaces in terms of the approximability of the corresponding algebras by negative (uniformly effectively separated) algebras. We compare negative and positive algebra representations from the standpoint of the structure of the corresponding effective spaces. For effective infinite topological spaces, we prove the existence of their infinite effective compact extensions. [ABSTRACT FROM AUTHOR]

The Hochschild cohomology algebra for the algebras of dihedral type in the subfamily of the family D(2ℬ), for which the parameter c is equal to 0, are described. The calculation of multiplication in this cohomology algebra, uses the minimal bimodule projective resolution for algebras under consideration, that was constructed in the previous paper of the authors. The obtained results allow to describe the Hochschild cohomology algebra also for algebras with c = 0 in the family D(2 A ). [ABSTRACT FROM AUTHOR]

In this paper, we consider the generalized Hilbert symbol in a higher local field of charactersitic 0 with first residue field of characteristic 0 as well and with perfect last residue field of positive characteristic p, which originates from higher local p-class field theory developed by I. Fesenko. Using the descent to a subfield of mixed characteristic, from the generalized Artin-Hasse and Iwasawa formulas proved in a previous paper we deduce respective Artin-Hasse and Iwasawa explicit reciprocity laws in the case under consideration. Bibliography: 6 titles. [ABSTRACT FROM AUTHOR]

The description of the subgroups of the orthogonal linear groups SO(2l,K) and GO(2l,K) over the field of fractions K of a local principal ideal domain R that contain the maximal split torus T = T(2l,R) with entries from R is presented. A similar result for overgroups of T in the case of a semilocal ring and a field was obtained earlier by N. Vavilov. The result of the present paper generalizes some known results as well. Bibliography: 15 titles. [ABSTRACT FROM AUTHOR]

The paper discusses the questions of coincidence of the basic nil-radicals on classes of algebraic Lie algebras and proves the local finite-dimensionality of Lie algebras with an algebraic adjoint representation of bounded degree over fields of sufficiently large positive characteristic., 1. Introduction The present paper supplements the results of A. I. Kostrikin and E. I. Zel'manov obtained during the solution of the restricted Burnside problem for groups of single prime [...]

For a pair of smooth transversally intersecting submanifolds in some enveloping smooth manifold, we study the algebra generated by pseudodifferential operators and (co)boundary operators corresponding to submanifolds. We establish that such an algebra has 18 types of generating elements. For operators from this algebra, we define the concept of symbol and obtain the composition formula. [ABSTRACT FROM AUTHOR]

The paper studies two numerical characteristics of matrix incidence algebras over finite fields associated with generating sets of such algebras: the minimal cardinality of a generating set and the length of an algebra. Generating sets are understood in the usual sense, the identity of the algebra being considered a word of length 0 in generators, and also in the strict sense, where this assumption is not used. A criterion for a subset to generate an incidence algebra in the strict sense is obtained. For all matrix incidence algebras, the minimum cardinality of a generating set and a generating set in the strict sense are determined as functions of the field cardinality and the order of the matrices. Some new results on the lengths of such algebras are obtained. In particular, the length of the algebra of "almost" diagonal matrices is determined, and a new upper bound for the length of an arbitrary matrix incidence algebra is obtained. [ABSTRACT FROM AUTHOR]

The paper studies the anticommutativity condition for elements of arbitrary real Cayley–Dickson algebras. As a consequence, the anticommutativity graphs on equivalence classes of such algebras are classified. Under some additional assumptions on the algebras considered, an expression for the centralizer of an element in terms of its orthogonalizer is obtained. Conditions sufficient for this interrelation to hold are provided. Also examples of real Cayley–Dickson algebras in which the centralizer and orthogonalizer of an element are not interrelated in this way are considered. [ABSTRACT FROM AUTHOR]

The Hochschild cohomology groups for algebras of dihedral type which are contained in the family D(3R) (from the famous K.Erdmann's classification) are calculated. In the calculation, a construction of the minimal bimodule resolution for algebras from the family under discussion, that is defined in the present paper, is used. [ABSTRACT FROM AUTHOR]

The Hochschild cohomology groups for algebras of semidihedral type, that are contained in the family SD(2B)1 (from the famous K. Erdmann's classification), are computed. In the calculation, a construction of the minimal bimodule resolution for algebras from the family under discussion, that is defined in the present paper, is used. [ABSTRACT FROM AUTHOR]

According to a result claimed by Pogorza_ly, selfinjective special biserial algebras can be stably equivalent to stably biserial algebras only, and these two classes coincide. By an example of Ariki, Iijima, and Park, the classes of stably biserial and selfinjective special biserial algebras do not coincide. In these notes based on some ideas from the Pogorzały paper, a detailed proof is given for the fact that a selfinjective special biserial algebra can be stably equivalent to a stably biserial algebra only. The structure of symmetric stably biserial algebras is analyzed. It is shown that in characteristic other than 2, the classes of symmetric special biserial (Brauer graph) algebras and symmetric stably biserial algebras coincide. Also a proof of the Auslander–Reiten conjecture for special biserial algebras is given. [ABSTRACT FROM AUTHOR]

In this paper, we review the results on topological ∗-algebras S(M), S(M, τ), and LS(M) of measurable, τ -measurable, and locally measurable operators affiliated with the von Neumann algebra M. Also, we consider relations between those algebras for different classes of von Neumann algebras and establish the continuity of operator-valued functions with respect to the local convergence in measure. We describe maximal commutative ∗-subalgebras of the algebra LS(M) as well. [ABSTRACT FROM AUTHOR]

This paper presents a simple way for diagonalization of matrices in an arbitrary field, with which one can calculate functions of matrices. [ABSTRACT FROM AUTHOR]