Groups, rings, and algebras: common strategic approaches and mutual influences

Authors

  • M. Dixon University of Alabama, Tuscaloosa, USA
  • M. Semko State Tax University, Irpin, Kyiv region
  • I. Subbotin National University, Los Angeles, USA

DOI:

https://doi.org/10.3842/umzh.v78i1-2.8953

Keywords:

algebra, brace, group, ring, module

Abstract

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.

References

1. L. A. Kurdachenko, O. O. Pypka, I. Ya. Subbotin, General theory of Leibniz algebras, Springer, Switzerland (2024); DOI: 10.1007/978-3-031-58148-9.

2. M. R. Dixon, L. A. Kurdachenko, I. Ya. Subbotin, Infinite groups: a roadmap to selected classical areas, Chapman and Hall/CRC, New York (2023); DOI: 10.1201/9781003147572.

3. M. R. Dixon, L. A. Kurdachenko, I. Ya. Subbotin, Linear groups: the accent on infinite dimensionality, Chapman and Hall/CRC, New York (2020); DOI: 10.1201/9781351008044.

4. M. R. Dixon, L. A. Kurdachenko, I. Ya. Subbotin, Ranks of groups: the tools, characteristics, and restrictions, Wiley, New Jersey (2017).

5. L. A. Kurdachenko, N. N. Semko, I. Ya. Subbotin, Insight into modules over Dedekind domains, Institute of Mathematics, National Academy of Sciences of Ukraine, Kiev (2008).

6. L. A. Kurdachenko, J. Otal, I. Ya. Subbotin, Artinian modules over group rings, Birkhäuser, Basel (2007); DOI: 10.1007/978-3-7643-7765-6.

7. L. A. Kurdachenko, J. Otal, I. Ya. Subbotin, Groups with prescribed quotient groups and associated module theory, World Scientific, New Jersey (2002); DOI: 10.1142/4839.

8. Д. И. Зайцев, Л. А. Курдаченко, А. В. Тушев, Модули над нильпотентными группами конечного ранга, Алгебра и логика, 24, № 6, 631--666 (1985).

9. M. J. Karbe, L. A. Kurdachenko, Just infinite modules over locally soluble groups, Arch. Math., 51, № 5, 401--411 (1988); DOI: 10.1007/BF01198623.

10. L. A. Kurdachenko, A. V. Tushev, D. I. Zaitcev, Noetherian modules over nilpotent groups of finite rank, Arch. Math., 56, № 5, 433--436 (1991); DOI: 10.1007/BF01200085.

11. Д. И. Зайцев, Л. А. Курдаченко, Группы с условием максимальности для неабелевых подгрупп, Укр. мат. журн., 43, № 7-8, 925--930 (1991).

12. Л. С. Казарин, Л. А. Курдаченко, Условия конечности и факторизация в бесконечных группах, Успехи мат. наук, 47, № 3, 75--114 (1992).

13. Л. А. Курдаченко, Артиновы модули над группами конечного ранга и группы с некоторыми условиями конечности, Бесконечные группы и примыкающие алгебраические структуры, Институт математики АН Украины, 144--159 (1993).

14. L. A. Kurdachenko, Artinian modules over groups of finite rank and the weak minimal condition for normal subgroups, Ric. Mat., 44, № 2, 303--335 (1995).

15. S. Franciosi, F. de Giovanni, L. A. Kurdachenko, The Schur property and groups with uniform conjugacy classes, J. Algebra, 174, № 3, 823--847 (1995); DOI: 10.1006/jabr.1995.1155.

16. L. A. Kurdachenko, B. V. Petrenko, I. Ya. Subbotin, On generalized hypercenters in Artinian modules, Commun. Algebra, 25, № 4, 1023--1046 (1997); DOI: 10.1080/00927879708825907.

17. Z. Y. Duan, L. A. Kurdachenko, I. Ya. Subbotin, Artinian modules over hyperfinite groups, Southeast Asian Bull. Math., 22, № 3, 273--286 (1998).

18. Л. А. Курдаченко, Модули над групповыми кольцами с некоторыми условиями конечности, Укр. мат. журн., 54, № 7, 931--940 (2002).

19. L. A. Kurdachenko, J. Otal, I. Ya. Subbotin, Baer-like decomposition of modules, Illinois J. Math., 47, № 1-2, 329--343 (2003); DOI: 10.1215/ijm/1258488159.

20. L. A. Kurdachenko, I. Ya. Subbotin, On some types of noetherian modules, J. Algebra Appl., 3, № 2, 169--179 (2004); DOI: 10.1142/S0219498804000812.

21. L. A. Kurdachenko, N. N. Semko, I. Ya. Subbotin, Artinian modules over FC-hypercentral groups, J. Algebra Appl., 6, № 3, 527--551 (2007); DOI: 10.1142/S0219498807002387.

22. L. A. Kurdachenko, I. Ya. Subbotin, V. A. Chupordya, On bounded Artinian finitary modules, Internat. J. Algebra Comput., 17, № 4, 881--893 (2007); DOI: 10.1142/S021819670700386x.

23. L. A. Kurdachenko, I. Ya. Subbotin, V. A. Chupordya, On the structure of some modules over generalized soluble groups, Turkish J. Math., 38, № 1, 52--59 (2014); DOI: 10.3906/mat-1302-47.

24. L. A. Kurdachenko, J. Otal, I. Ya. Subbotin, On the structure of some noetherian modules over group rings, Internat. J. Algebra Comput., 24, № 2, 233--249 (2014); DOI: 10.1142/s0218196714500131.

25. L. A. Kurdachenko, I. Ya. Subbotin, V. A. Chupordya, On modules with few minimax cocentralizers, Commun. Algebra, 43, № 2, 871--885 (2015); DOI: 10.1080/00927872.2013.855069.

26. L. A. Kurdachenko, I. Ya. Subbotin, V. A. Chupordya, On the relations between the central factor-module and the derived submodule in modules over groups rings, Comment. Math. Univ. Carol., 56, № 4, 433--445 (2015); DOI: 10.14712/1213-7243.2015.136.

27. L. A. Kurdachenko, M. Trombetti, Modules over group rings with restrictions on some factor-modules, Mediterr. J. Math., 14, № 3, 135 (2017); DOI: 10.1007/s00009-017-0937-3.

28. L. A. Kurdachenko, M. Trombetti, Just non-Artinian modules over some group rings, Turkish J. Math., 42, № 2, 1242--1254 (2018); DOI: 10.3906/mat-1704-4.

29. M. R. Dixon, L. A. Kurdachenko, I. Ya. Subbotin, On composition factors in modules over some group rings, Turkish J. Math., 43, № 6, 2806--2817 (2019); DOI: 10.3906/mat-1908-65.

30. L. A. Kurdachenko, V. A. Bovdi, Some ranks of modules over group rings, Commun. Algebra, 49, № 2, 1225--1234 (2020); DOI: 10.1080/00927872.2020.1831007.

31. L. A. Kurdachenko, V. A. Bovdi, Modules over some group rings, having d-generator property, Ric. Mat., 71, № 1, 135--145 (2022); DOI: 10.1007/s11587-021-00581-5.

32. Л. А. Курдаченко, В. Э. Горецкий, В. В. Пылаев, Группы с некоторой системой минимаксных фактор-групп, Допов. АН України, Сер. А, № 3, 17--20 (1988).

33. Л. А. Курдаченко, В. В. Пылаев, О группах с минимаксными фактор-группами, Укр. мат. журн., 42, № 5, 620--625 (1990).

34. S. Franciosi, F. de Giovanni, L. A. Kurdachenko, Groups whose proper quotients are FC-groups, J. Algebra, 186, № 2, 544--577 (1996); DOI: 10.1006/jabr.1996.0386.

35. L. A. Kurdachenko, I. Ya. Subbotin, Groups whose proper quotients are hypercentral, J. Aust. Math. Soc., 65, № 2, 224--237 (1998); DOI: 10.1017/s1446788700034972.

36. L. A. Kurdachenko, J. Otal, Some noetherian modules and non-monolithic just non-CC-groups, J. Group Theory, 2, № 1, 53--64 (1999).

37. L. A. Kurdachenko, P. Soules, Just non-SRI*-groups, Proceedings of the Second Panhellenic Conference in Algebra and Number Theory, Bull. Greek Math. Soc., 42, 33--42 (1999).

38. L. A. Kurdachenko, I. Ya. Subbotin, Groups with many periodic factor-groups, Commun. Algebra, 28, № 3, 1593--1602 (2000).

39. L. A. Kurdachenko, J. Otal, Simple modules over CC-groups and monolithic just non-CC-groups, Boll. Unione Mat. Ital., 4-B, № 8, 381--390 (2001).

40. L. A. Kurdachenko, J. Otal, Groups with proper hypercyclic homomorphic images, Ric. Mat., 50, № 1, 53--65(2001).

41. L. A. Kurdachenko, J. Otal, FC-groups all of whose factor groups are residually finite, Commun. Algebra, 31, № 3, 1235--1251(2003); DOI: 10.1081/AGB-120017764.

42. L. A. Kurdachenko, I. Ya. Subbotin, On some infinite dimensional linear groups, Commun. Algebra, 29, № 2, 519--527 (2001); DOI: 10.1081/AGB-100001521.

43. L. A. Kurdachenko, I. Ya. Subbotin, On some infinite dimensional linear groups, Southeast Asian Bull. Math., 26, 773--787 (2002).

44. L. A. Kurdachenko, I. Ya. Subbotin, On some infinite dimensional linear group, Groups St. Andrews 2001 in Oxford, vol. II, London Math. Soc. Lecture Note Ser., 305, Cambridge Univ. Press (2003), pp. 377--384.

45. M. R. Dixon, M. J. Evans, L. A. Kurdachenko, Linear groups with the minimal condition on subgroups of infinite central dimension, J. Algebra, 277, № 1, 172--186 (2004); DOI: 10.1016/j.jalgebra.2004.02.09.

46. L. A. Kurdachenko, I. Ya. Subbotin, Linear groups with the maximal condition on subgroups of infinite central dimension, Publ. Mat., 50, № 1, 103--131 (2006); DOI: 10.5565/publmat_50106_06.

47. M. R. Dixon, L. A. Kurdachenko, Linear groups with infinite central dimension, Groups St. Andrews 2005, vol. I, London Math. Soc. Lecture Note Ser., 339, Cambridge Univ. Press (2007), pp. 306--312.

48. O. Yu. Dashkova, M. R. Dixon, L. A. Kurdachenko, Linear groups with rank restrictions on the subgroups of infinite central dimension, J. Pure Appl. Algebra, 208, № 3, 785--795 (2007); DOI: 10.1016/j.jpaa.2006.04.002.

49. L. A. Kurdachenko, J.-M. Muňos-Escolano, J. Otal, Locally nilpotent linear groups with the weak chain conditions on subgroups of infinite central dimension, Publ. Mat., 52, № 1, 151--169 (2008); DOI: 10.5565/PUBLMAT_52108_07.

50. L. A. Kurdachenko, J.-M. Muňos-Escolano, J. Otal, Antifinitary linear groups, Forum Math., 20, № 1, 27--44 (2008); DOI: 10.1515/FORUM.2008.002.

51. L. A. Kurdachenko, J.-M. Muňos-Escolano, J. Otal, N. N. Semko, Locally nilpotent linear groups with restrictions on their subgroups of infinite central dimension, Geom. Dedicata, 138, № 1, 69--81 (2009); DOI: 10.1007/s10711-008-9299-0.

52. L. A. Kurdachenko, J.-M. Muňos-Escolano, J. Otal, Soluble linear groups with some restrictions on subgroups of infinite central dimension, Ischia Group Theory 2008, Proceedings of the Conference in Group Theory, World Sci. Publ., Hackensack, NJ (2009), pp. 156--173; DOI: 10.1142/9789814277808_0012.

53. M. R. Dixon, L. A. Kurdachenko, J. Otal, On some infinite dimensional linear groups, Note Mat., 30, № 1, 21--36 (2010).

54. M. R. Dixon, L. A. Kurdachenko, J. Otal, Linear groups with bounded action, Algebra Colloq., 18, № 3, 487--498 (2011); DOI: 10.1142/s1005386711000368.

55. M. R. Dixon, L. A. Kurdachenko, J.-M. Muňos-Escolano, J. Otal, Trends in infinite dimensional linear groups, Groups St. Andrews 2009 in Bath. Vol. I, London Math. Soc. Lecture Note Ser., 387, Cambridge Univ. Press (2011), pp. 271--282.

56. M. R. Dixon, L. A. Kurdachenko, J. Otal, Linear groups with finite dimensional orbits, Ischia Group Theory 2010, Proceedings of the Conference in Group Theory, World Scientific (2012), pp. 131--145.

57. L. A. Kurdachenko, J.-M. Muňos-Escolano, J. Otal, Groups acting on vector spaces with a large family of invariant subspaces, Linear Multilinear Algebra, 60, № 4, 487--498 (2012).

58. M. R. Dixon, L. A. Kurdachenko, J. Otal, Linear analogues of theorems of Schur, Baer and Hall, Int. J. Group Theory, 2, № 1, 79--89 (2013).

59. M. R. Dixon, L. A. Kurdachenko, J. Otal, On the structure of some infinite dimensional linear groups}. Commun. Algebra, 45, № 1, 234--246 (2017); DOI: 10.1080/00927872.2016.1175593.

60. M. R. Dixon, L. A. Kurdachenko, I. Ya. Subbotin, On an analogue of a theorem of P. Hall for infinite dimensional linear groups, Eur. J. Math., 6, № 2, 577--589 (2020); DOI: 10.1007/s40879-019-00334-7.

61. M. R. Dixon, L. A. Kurdachenko, I. Ya. Subbotin, Some linear groups with G-core-free subspaces, Commun. Algebra, 48, № 12, 5455--5464 (2020); DOI: 10.1080/00927872.2020.1790583.

62. L. A. Kurdachenko, I. Ya. Subbotin, O. O. Pypka, On some relationships between the factors of upper and lower central series in Lie algebras, Serdica Math. J., 41, 293--306 (2015).

63. L. A. Kurdachenko, J. Otal, O. O. Pypka, Relationships between factors of canonical central series of Leibniz algebras, Eur. J. Math., 2, № 2, 565--577 (2016); DOI: 10.1007/S40879-016-0093-5.

64. V. A. Chupordya, L. A. Kurdachenko, I. Ya. Subbotin, On some ``minimal'' Leibniz algebras, J. Algebra Appl., 16, № 5, 1750082 (2017); DOI: 10.1142/S0219498817500827.

65. L. A. Kurdachenko, N. N. Semko, I. Ya. Subbotin, The Leibniz algebras whose subalgebras are ideal, Open Math., 15, № 1, 92--100 (2017); DOI: 10.1515/math-2017-0010.

66. L. A. Kurdachenko, M. M. Semko, I. Ya. Subbotin, On the anticommutativity in Leibniz algebras, Algebra Discrete Math., 26, № 1, 97-109 (2018).

67. L. A. Kurdachenko, I. Ya. Subbotin, V. S. Yashchuk, Leibniz algebras whose subideals are ideals, J. Algebra Appl., 17, Article 1850151 (2018); DOI: 10.1142/S0219498818501517.

68. L. A. Kurdachenko, J. Otal, I. Ya. Subbotin, On some properties of the upper central series in Leibniz algebras, Comment. Math. Univ. Carol., 60, № 2, 161--175 (2019); DOI:10.14712/1213-7243.2019.009.

69. L. A. Kurdachenko, I. Ya. Subbotin, V. S. Yashchuk, Leibniz algebras whose subalgebras are left ideals, Serdica Math. J., 46, 175--194 (2020).

70. V. A. Chupordya, L. A. Kurdachenko, N. N. Semko, On the structure of Leibniz algebras whose subalgebras are ideals or core-free, Algebra Discrete Math., 29, № 2, 180--194 (2020); DOI: 10.12958/adm1533.

71. L. A. Kurdachenko, N. N. Semko, I. Ya. Subbotin, Applying group theory philosophy to Leibniz algebras: some new developments, Adv. Group Theory Appl., 9, 71--121 (2020); DOI: 10.32037/agta-2020-004.

72. L. A. Kurdachenko, I. Ya. Subbotin, V. S. Yashchuk, Some antipodes of ideals in Leibniz algebras, J. Algebra Appl., 19, № 6, 2050113 (2020); DOI: 10.1142/S0219498820501134.

73. V. A. Chupordya, L. A. Kurdachenko, N. N. Semko, On the structure of Leibniz algebras, whose subalgebras are ideals or core-free, Dopov. Nac. Akad. Nauk Ukr., № 7, 17--21 (2020); DOI: 10.15407/dopovidi2020.07.017.

74. L. A. Kurdachenko, O. O. Pypka, I. Ya. Subbotin, On Leibniz algebras whose subalgebras are either ideals or self-idealizing subalgebras}. Ukr. Math. J., 73, № 6, 944--962 (2021); DOI: 10.1007/s11253-021-01969-0.

75. L. A. Kurdachenko, O. O. Pypka, I. Ya. Subbotin, On the automorphism groups of some Leibniz algebras, Int. J. Group Theory, 12, № 1, 1--20 (2023); DOI: 10.22108/ijgt.2021.130057.1735.

76. L. A. Kurdachenko, I. Ya. Subbotin, V. S. Yashchuk, On the endomorphisms and derivations of some Leibniz algebras, J. Algebra Appl., 23, № 1, 2450002 (2024); DOI: 10.1142/S0219498824500026.

77. L. A. Kurdachenko, I. Ya. Subbotin, On the structure of some one-generator braces, Proc. Edinb. Math. Soc. (2), 67, № 2, 566--576 (2024); DOI: 10.1017/S0013091524000154.

68. A. Ballester-Bolinches, R. Esteban-Romero, L. A. Kurdachenko, V. P'erez-Calabuig, On the structure of left braces satisfying the minimal condition for subbraces, J. Algebra, 662, № 2, 528--544 (2025); DOI: 10.1016/j.jalgebra.2024.08.011.

79. A. Ballester-Bolinches, R. Esteban-Romero, L. A. Kurdachenko, V. P'erez-Calabuig, On the structure of some left braces, Int. J. Group Theory, 14, № 2, 47--58 (2025); DOI: 10.22108/ijgt.2023.139145.1872.

80. M. R. Dixon, L. A. Kurdachenko, I. Ya. Subbotin, Generalized nilpotent braces and nilpotent groups, Int. J. Group Theory, 14, № 1, 25--45 (2025); DOI: 10.22108/ijgt.2023.138490.1860.

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26.01.2026

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Vol. 78 No. 1-2 (2026)

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Research articles

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Copyright (c) 2026 M. Dixon, M. Semko, I. Subbotin

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How to Cite

Dixon, M., et al. “Groups, Rings, and Algebras: Common Strategic Approaches and Mutual Influences”. Ukrains’kyi Matematychnyi Zhurnal, vol. 78, no. 1-2, Jan. 2026, pp. 38–46, https://doi.org/10.3842/umzh.v78i1-2.8953.