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Characteristics of partially pseudo-ordered (K-ordered) rings are considered. Properties of ideals in pseudo-ordered rings are described. A variation of the concept of the prime radical for a ring in the subclass of directed pseudo-ordered rings (AO-ordered rings) is investigated. The description of elements of AO-prime radicals for AO-ordered rings is obtained.

In this review, we consider applications of nonassociative algebraic structures for the construction of linearly optimal codes and cryptosystems.

This paper is devoted to the application aspects of homomorphic cryptography. It provides a description of a fully homomorphic matrix polynomial-based encryption scheme. It also gives the results of practical comparison of fully homomorphic encryption schemes. We consider some special cases of homomorphic encryption allowing computations of a limited number of functions.

A variety of linear algebras is said to be Schreier if any subalgebra of a free algebra of this variety is free. A system of elements of a free algebra is primitive if there is a complement of this system with respect to a free generating set of the free algebra. An element of a free algebra of a Schreier variety is said to be almost primitive if it is not primitive in the free algebra, but it is a primitive element of any subalgebra that contains it. This survey article is devoted to the study of primitive and almost primitive elements of Schreier varieties.

In this paper, we prove a criterion for elementary equivalence of stable linear groups over commutative local rings with invertible two.

We analyze algorithms for open construction of a key on some noncommutative group. Algorithms of factorization and decomposition for associative algebras (of small dimension) are considered. A survey of applications (in particular, in cryptography) of so-called “hidden matrices” is given.

In this paper, main properties of a commutator of two normal subloops of a loop are considered. The notion of a prime radical of loops is introduced and its characterization as a set of strongly Engel elements is given. Also an Ω-prime radical of Ω-loops is defined and its elementwise characterization is given.

The remarkable class of measurable functions was introduced by classics of function theory. It has found different applications in various branches of mathematics. However this class turned out too restrictive for solving some natural mathematical problems because it is essentially connected with the property of countability. Therefore, along with it another remarkable class, essentially connected with the property of finiteness, was introduced. It is the class of uniform functions. Measurable functions are described both in the classical Lebesgue–Borel language of preimages and in the quite new language of covers. Uniform functions are described in the language of covers exclusively. Both the families of measurable functions and the families of uniform functions are determined by the rigid structure of their supports (descriptive spaces). For this reason, mathematicians weakened more that once the rigidity of the structure of descriptive spaces at the expense of using the additional property of negligence. The present paper is devoted to a contemporary formalization of the indicated ideas. Some applications of the introduced classes of functions to solving a number of known mathematical problems is traced in the paper.

In this paper we prove that for a linearly ordered skewfield the groups of quitients of the semigroup G n (D) coincides with the group GL n (D) for n ≥ 3.

This paper contains a review of results on graded quotient rings and new results proved by the authors.

The problem of characterization of integrals as linear functionals is considered in the paper. It starts from the familiar results of F. Riesz (1909) and J. Radon (1913) on integral representation of bounded linear functionals by Riemann–Stieltjes integrals on a segment and by Lebesgue integrals on a compact in \( {\mathbb{R}^n} \), respectively. After works of J. Radon, M. Frechet, and F. Hausdorff the problem of characterization of integrals as linear functionals took the particular form of the problem of extension of Radon’s theorem from \( {\mathbb{R}^n} \) to more general topological spaces with Radon measures. This problem has turned out difficult and its solution has a long and rich history. Therefore, it may be naturally called the Riesz–Radon–Frechet problem of characterization of integrals. The important stages of its solution are connected with such mathematicians as S. Banach, S. Saks, S. Kakutani, P. Halmos, E. Hewitt, R. E. Edwards, N. Bourbaki, V. K. Zakharov, A. V. Mikhalev, et al. In this paper, the Riesz–Radon–Fr´echet problem is solved for the general case of arbitrary Radon measures on Hausdorff spaces. The solution is given in the form of a general parametric theorem in terms of a new notion of the boundedness index of a functional. The theorem implies as particular cases well-known results of the indicated authors characterizing Radon integrals for various classes of Radon measures and topological spaces.

In this paper, topologically primitive rings and rings possessing a faithful topologically irreducible module and bounded by this module are considered for the investigation of properties of their topological Jacobson radical. We investigate the topological Jacobson radical in some classes of topological rings such as left topologically Artinian rings, topological rings possessing a basis of neighborhoods of zero consisting of ideals, compact rings, and bounded strictly linearly compact rings.

In this paper, we give a simpler proof of the Golubchik–Mikhalev–Zelmanov theorem on the structure of isomorphisms between general linear groups over associative rings, and also prove an extension of this theorem for linear groups over rings graded by an Abelian group.

We obtain criteria that answer the question of when an anti-isomorphism of graded endomorphism rings of the strict gr-generators is induced by a graded Morita anti-equivalence or a graded anti-semilinear isomorphism.

This review paper is devoted to the Jacobi bound for systems of partial differential polynomials. We prove the conjecture for the system of n partial differential equations in n differential variables which are independent over a prime differential ideal \(\mathfrak{p}\). On the one hand, this generalizes our result about the Jacobi bound for ordinary differential polynomials independent over a prime differential ideal \(\mathfrak{p}\) and, on the other hand, the result by Tomasovic, who proved the Jacobi bound for linear partial differential polynomials.

Improved algorithms to construct complements of primitive systems of elements of free nonassociative algebras with respect to free generating sets and algorithms to realize the rank of a system of elements are constructed and implemented.

The notion of the prime radical for a directed group is considered. A series of results on commutants of AO-groups is obtained. Prime radicals of AO-groups are described.

The aim of this article is to consider the theoretical basis of an automated-system-vulnerability scanning (assessment) process and to prove the possibility of determining the internal-state distribution based on conducting the scanning process. According to international information security standards, the scanning process is called the vulnerability assessment process. Conducting this process and the existence of the required technical resources is a necessary condition for creating effective information-security systems.

The authors investigate semirings of cyclic types from the algebraic point of view. To simplify and facilitate the analysis, the local Fourier transform of these semirings is introduced. The authors describe zero divisors, nilpotent elements, invertible elements, idempotents, and the Jacobson radical.

In this paper, we prove that the semigroups of invertible matrices with nonnegative elements over linearly ordered associative rings are elementarily equivalent if and only if the matrices have the same dimension and the rings are elementarily equivalent as ordered rings.

The concept of prime radicals is important in the study of rings and groups. The purpose of this paper is to investigate a generalization of this concept to directed groups. Some results are obtained concerning convex directed subgroups of AO-groups. Prime radicals of pl-groups are described.

In this paper, we introduce the class of graded Ω-groups, which includes: groups; associative, conformal and vertex algebras; Lie algebras and graded algebras. The graded prime radical of a graded Ω-group is defined, and its elementwise characterization is given. It is shown that the graded prime radical of a graded Ω-groups with a finiteness condition coincides with the lower weakly solvable (in the Parfyonov sense) radical.

The paper is aimed at the problem of lifting automorphisms and derivations of a perfect conformal Lie superalgebra to its central extension. Such lifting is possible with the help of a universal central extension whose tensor model is constructed here for this purpose.

In this paper, we prove that every automorphism of the semigroup of invertible matrices with nonnegative elements over a linearly ordered associative ring on some specially defined subgroup coincides with the composition of an inner automorphism of the semigroup, an order-preserving automorphism of the ring, and a central homothety.

In this paper, we study Steinberg unitary Lie conformal algebras, which are universal central extensions of unitary Lie conformal algebras. We describe the kernels of these extensions by means of skew-dihedral homology.

This work is devoted to results obtained in the model theory of regular polygons. We give a characterization of monoids with axiomatizable and model-complete class of regular polygons. We describe monoids with complete class of regular polygons that satisfy some additional conditions. We study monoids whose regular core is represented as a union of finitely many principal right ideals and all regular polygons over which have a stable and superstable theory. We prove the stability of the class of all regular polygons over a monoid provided this class is axiomatizable and model-complete. We also describe monoids for which the class of all regular polygons is superstable and ω-stable provided this class is axiomatizable and model-complete.

In this paper, we consider two approaches toward the definition of a topological prime radical of a topological group. In the first approach, the prime quasi-radical η(G) is defined as the intersection of all closed prime normal subgroups of a topological group G. Its properties are investigated. In the second approach, we consider the set η′(G) of all topologically strictly Engel elements of a topological group G. Its properties are investigated. It is proved that η′(G) is a radical in the class of all topological groups possessing a basis of neighborhoods of the identity element consisting of normal subgroups.

In this paper, we consider a topological prime quasi-radical μ(R), which is the intersection of closed prime ideals in a topological ring R. Examples are given that show that μ(R) is different from those topological analogs of the prime radical that have been studied earlier. The topological prime quasi-radicals of matrix rings and rings of polynomials are investigated.

In this work we study central extensions of Lie superalgebras g(A), where g = P(n − 1) are simple Lie superalgebras from one of the so-called “strange” series. We show that universal central extensions of these Lie superalgebras ara trivial, i.e., they are isomorphic to the superalgebras g(A).

This paper is a review of some recent results of elementary equivalence of linear and algebraic groups and our recent new results of elementary equivalence of categories of modules, endomorphism rings of modules, and automorphism groups of modules.

The interrelations between the theory of linear transformations preserving matrix invariants and different branches of mathematics are surveyed here. The preferences are given for those methods and motivations to study these transformations that arise from general algebra.

The main purpose of this paper is to prove three density theorems for rings graded by semigroups and modules graded by acts over these semigroups with some cancellation conditions. In addition, the density theorem for superrings and supermodules is proved.

In this article the universal central extensions of Lie superalgebras related to simple complex Lie superalgebras having nondegenerate Killing form are considered.

The crisis that arose in naive set theory at the beginning of 20th century led to the appearance of several precise axiomatic constructions of theories of mathematical totalities. Among them, set theory in Zermelo–Frankel’s axiomatics (ZF) (see [1]) and the class theory and set theory in the axiomatics of Neumann–Bernays–Goedel (NBG) (see [2] and [3]) are the most widely used. These axiomatic theories remove all known paradoxes of naive set theory at the expense of a sharp restriction of possible representational tools. At the same time, they give the possibility of involving almost all mathematical objects and constructions existing at that time in the framework of these theories. However, even in 1945, in the fundamental work [4], Eilenberg and MacLane introduced a new mathematical concept of a category. Since that time, category theory has become an independent field of mathematics. However, from the very beginning, category theory encoutered the unpleasant fact that it cannot be included in the framework of the axiomatic set theory ZF and that of the axiomatic theory of classes and sets NBG (see [4]). Therefore, MacLane in [5], written in 1959, has posed the general problem of constructing a new and more flexible axiomatic set theory which can serve as an adequate logical foundation of the whole naive category theory. Variants of new axiomatic theories of mathematical totalities adopted for one or another need of category theory were suggested by Ehresmann [6], Dedecker [7], Sonner [8], Grotendieck [9], da Costa [10] and [11], Isbell [12], MacLane [13] and [14], Fefferman [15], Herrlich and Strecker [16], and others. The axiomatics of Ehresmann–Dedecker and Sonner–Grotendieck contains a very strong axiom in addition to the axioms of ZF, which asserts that every set is contained in a certain universal set. In the axiomatics of MacLane, it is assumed that there exists at least one universal set. Versions close to them were suggested by Isbell and Feferman. The axiomatics of Herlich–Strecker deals with objects of the following three types: sets, classes, and conglomerates. Within the framework of each of these axiomatic theories, some definitions of a category and a functor were given. However, the concept of a category given in [12–16] is not closed under important operations of naive set theory such as the “category of categories” and the “category of functors” (see [17, 8.4]). Within the framework of the axiomatic theories in [6–9], the definitions of a U -category and a U functor which consist of elements of a universal set U were given. Such a concept of a category is closed under operations such as the V-category of U-categories and the V-category of U -functors, where V is a certain universal set containing the universal set U . However, U -categories can contain objects of a bounded cardinality, which is not natural from the point of view of naive category theory. Therefore, in [13], MacLane noted the following about this approach: “This stronger assumption evidently provides for each universe U a category of all those groups which are members of U . However, this does not provide any category of all groups.” The second axiomatics of da Costa in [10] and [11] has the same shortcoming. His first axiomatics in [10] and [11] is free from this shortcoming, and, at the same time, it is closed with respect to the operation of naive category theory mentioned above. Therefore, it is the most preferable among all the