Gr\"obner-Shirshov bases for Lie algebras over a commutative algebra

In this paper we establish a Gr\"{o}bner-Shirshov bases theory for Lie algebras over commutative rings. As applications we give some new examples of special Lie algebras (those embeddable in associative algebras over the same ring) and non-special Lie algebras (following a suggestion of P.M. Cohn (1963) \cite{Conh}). In particular, Cohn's Lie algebras over the characteristic $p$ are non-special when $p=2,\ 3,\ 5$. We present an algorithm that one can check for any $p$, whether Cohn's Lie algebras is non-special. Also we prove that any finitely or countably generated Lie algebra is embeddable in a two-generated Lie algebra

Lyndon-Shirshov basis and anti-commutative algebras

Chen, Fox, Lyndon 1958 \cite{CFL58} and Shirshov 1958 \cite{Sh58} introduced non-associative Lyndon-Shirshov words and proved that they form a linear basis of a free Lie algebra, independently. In this paper we give another approach to definition of Lyndon-Shirshov basis, i.e., we find an anti-commutative Gr\"{o}bner-Shirshov basis $S$ of a free Lie algebra such that $Irr(S)$ is the set of all non-associative Lyndon-Shirshov words, where $Irr(S)$ is the set of all monomials of $N(X)$, a basis of the free anti-commutative algebra on $X$, not containing maximal monomials of polynomials from $S$. Following from Shirshov's anti-commutative Gr\"{o}bner-Shirshov bases theory \cite{S62a2}, the set $Irr(S)$ is a linear basis of a free Lie algebra

The digitalization features of the Russian social media market insurance service

Purpose: The article aims to identify the features of digital techniques introduction and functioning in the insurance industry and the social insurance system. The information techniques used allows to quickly and with the best results to process large amounts of data, thereby increasing the efficiency of all the spheres to reduce social and commercial risks. Design/Methodology/Approach: In order to substantiate introducing the digital techniques expediency in the activity of the insurance system, it is necessary, first, to consider the advantages and disadvantages of information techniques use in the commercial and social insurance. Second, to describe the main digital programs, the implementation of which will increase the targeting and personalization of insurance services. Findings: For the digital techniques introduction in the sphere of insurance relations it is necessary to form the wholly new structure of insurance assets that meet the priorities of the digital economy; to create the necessary conditions for the development and implementation of modern actuarial techniques; to create conditions for increasing incomes and the life standard of the population in order to stimulate demand for insurance services; to change the structure and quality of social services. Practical implications: The results of the study can be implemented in the practice of social funds and insurance companies in order to improve the quality of insurance services. Originality/value: The main contribution of this research is to transfer the processes and mechanisms for the global digital economy and global digital space formation to the social and insurance relations.peer-reviewe

Spiritual and moral education of Russian soldiers

The defense of the Fatherland is always considered in Russia strictly prescribed. The basis of the military spirit is a spiritual and moral education of the soldiers. Service in the armed forces is dominated by the principle of statehood. The Russian Orthodox Church has played a leading role in spiritual and moral education of the Russian soldiers.Защита Отечества всегда считалась на Руси делом священным. Основой воинского духа является духовно-нравственное воспитание воинов. Служение в вооружённых силах проникнуто принципом государственности. Русская Православная Церковь играла ведущую роль в духовно-нравственном воспитании российских воинов

Anti-commutative Groebner-Shirshov basis of a free Lie algebra

One of the natural ways to prove that the Hall words (Philip Hall, 1933) consist of a basis of a free Lie algebra is a direct construction: to start with a linear space spanned by Hall words, to define the Lie product of Hall words, and then to check that the product yields the Lie identities (Marshall Hall, 1950). Here we suggest another way using the Composition-Diamond lemma for free anti-commutative (non-associative) algebras (A.I. Shirshov, 1962)

Composition-Diamond Lemma for Tensor Product of Free Algebras

In this paper, we establish Composition-Diamond lemma for tensor product $k< X> \otimes k$ of two free algebras over a field. As an application, we construct a Groebner-Shirshov basis in $k \otimes k$ by lifting a Groebner-Shirshov basis in $k[X] \otimes k$, where $k[X]$ is a commutative algebra.Comment: 19 page

Development trends for the insurance industry in Russia

The paper considers the main problems and trends for the insurance market in Russia. Threats and risks that may affect its further development are also presented in it. The work reveals the issues of improving the mechanisms of insurance protection of business entities and identifies priority types of insurance. The analysis of the current trends in the modern insurance market of Russia, important statistical indicators of the importance of insurance in the modern financial system are also provided in the research. Research of the insurance market and its problems is aimed at understanding the processes taking place in the insurance in order to improve the interaction of the state, insurance companies and consumers of insurance services.peer-reviewe