Groups, rings, and algebras: common strategic approaches and mutual influences
UDC 512.5 We focus on recent promising trends in the application of the key concepts and approaches from the classical infinite-group theory to various branches of algebra, such as modules over group rings, infinite-dimensional linear groups, Leibniz algebras, other generalizations of the Lie algebr...
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| Date: | 2026 |
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| Main Authors: | , , , , , |
| Format: | Article |
| Language: | Ukrainian |
| Published: |
Institute of Mathematics, NAS of Ukraine
2026
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/8953 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860513349053186048 |
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| author | Dixon, M. Semko, M. Subbotin, I. Діксон, Мартин Семко, Микола Субботін, Ігор |
| author_facet | Dixon, M. Semko, M. Subbotin, I. Діксон, Мартин Семко, Микола Субботін, Ігор |
| author_sort | Dixon, M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2026-03-21T11:04:14Z |
| description | UDC 512.5
We focus on recent promising trends in the application of the key concepts and approaches from the classical infinite-group theory to various branches of algebra, such as modules over group rings, infinite-dimensional linear groups, Leibniz algebras, other generalizations of the Lie algebras, and braces. The efficacy of these trends has been well-documented in a series of recent books from reputable publishers. In our article, we present a concise overview of these emerging trends. The analysis of the mutual influence of algebraic systems promotes deeper understanding of their individual and collective significance and illustrates the unity and diversity typical of contemporary mathematics. We believe that the subsequent development of investigations in this field would promote the appearance of new discoveries and innovations clarifying the fundamental role played by the groups, rings, algebras and other algebraic structures in modern mathematics. |
| doi_str_mv | 10.3842/umzh.v78i1-2.8953 |
| first_indexed | 2026-03-24T03:43:16Z |
| format | Article |
| fulltext |
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| id | umjimathkievua-article-8953 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian |
| last_indexed | 2026-03-24T03:43:16Z |
| publishDate | 2026 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/bd/89e43d1740ac0296cbafa553098e04bd |
| spelling | umjimathkievua-article-89532026-03-21T11:04:14Z Groups, rings, and algebras: common strategic approaches and mutual influences Групи, кільця та алгебри: загальні стратегічні підходи та взаємний вплив Dixon, M. Semko, M. Subbotin, I. Діксон, Мартин Семко, Микола Субботін, Ігор algebra brace group ring module алгебра брейс група кільце модуль UDC 512.5 We focus on recent promising trends in the application of the key concepts and approaches from the classical infinite-group theory to various branches of algebra, such as modules over group rings, infinite-dimensional linear groups, Leibniz algebras, other generalizations of the Lie algebras, and braces. The efficacy of these trends has been well-documented in a series of recent books from reputable publishers. In our article, we present a concise overview of these emerging trends. The analysis of the mutual influence of algebraic systems promotes deeper understanding of their individual and collective significance and illustrates the unity and diversity typical of contemporary mathematics. We believe that the subsequent development of investigations in this field would promote the appearance of new discoveries and innovations clarifying the fundamental role played by the groups, rings, algebras and other algebraic structures in modern mathematics. УДК 512.5 Висвітлено новітні перспективні тенденції у застосуванні ключових концепцій і підходів класичної теорії нескінченних груп до різних розділів алгебри, таких як модулі над груповими кільцями, нескінченновимірні лінійні групи, алгебри Лейбніца, інші узагальнення алгебр Лі та брейси. Ефективність таких підходів задокументовано у серії нещодавно опублікованих книг в авторитетних видавництвах. Подано стислий огляд цих тенденцій. Взаємний вплив алгебраїчних систем сприяє глибшому розумінню їхнього індивідуального та колективного значення, ілюструючи єдність і різноманіття, які характеризують сучасну математику. Ми вважаємо, що подальший розвиток досліджень у цій галузі сприятиме відкриттям і новаціям, підкреслюючи фундаментальну роль груп, кілець, алгебр та інших алгебраїчних структур у математиці. Institute of Mathematics, NAS of Ukraine 2026-03-02 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/8953 10.3842/umzh.v78i1-2.8953 Ukrains’kyi Matematychnyi Zhurnal; Vol. 78 No. 1-2 (2026); 38–46 Український математичний журнал; Том 78 № 1-2 (2026); 38–46 1027-3190 uk https://umj.imath.kiev.ua/index.php/umj/article/view/8953/10614 Copyright (c) 2026 M. Dixon, M. Semko, I. Subbotin |
| spellingShingle | Dixon, M. Semko, M. Subbotin, I. Діксон, Мартин Семко, Микола Субботін, Ігор Groups, rings, and algebras: common strategic approaches and mutual influences |
| title | Groups, rings, and algebras: common strategic approaches and mutual influences |
| title_alt | Групи, кільця та алгебри: загальні стратегічні підходи та взаємний вплив |
| title_full | Groups, rings, and algebras: common strategic approaches and mutual influences |
| title_fullStr | Groups, rings, and algebras: common strategic approaches and mutual influences |
| title_full_unstemmed | Groups, rings, and algebras: common strategic approaches and mutual influences |
| title_short | Groups, rings, and algebras: common strategic approaches and mutual influences |
| title_sort | groups, rings, and algebras: common strategic approaches and mutual influences |
| topic_facet | algebra brace group ring module алгебра брейс група кільце модуль |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/8953 |
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