Characterization and commuting probability of n-centralizer finite rings

• Dutta, Jutirekha ^[1] ; Basnet, Dhiren Kumar ^[1] ; Nath, Rajat Kanti ^[1]
  1. [1] Tezpur University

     Tezpur University

     India

     
• Localización: Proyecciones: Journal of Mathematics, ISSN 0716-0917, ISSN-e 0717-6279, Vol. 42, Nº. 6, 2023, págs. 1489-1498
• Idioma: inglés
• DOI: 10.22199/issn.0717-6279-5440
• Enlaces
  • Texto completo
• Resumen

  • Let R be a finite ring. The commuting probability of R is the probability that any two randomly chosen elements of R commute. A ring R is called an n-centralizer ring if it has n distinct centralizers. In this paper, we characterize some n-centralizer finite rings and compute their commuting probabilities.

• Referencias bibliográficas
  • A. Abdollahi, S. M. J. Amiri and A. M. Hassanabadi, “Groups with specific number of centralizers”, Houston J. Math., vol. 33, no. 1, pp. 43-57,...
  • A. R. Ashrafi, “On finite groups with a given number of centralizers”, Algebra Colloq., vol. 7, no. 2, pp. 139-146, 2000.
  • A. R. Ashrafi, “Counting the centralizers of some finite groups”, Korean J. Comput. Appl. Math., vol. 7, no. 1, pp. 115-124, 2000.
  • A. R. Ashrafi and B. Taeri, “On finite groups with exactly seven element centralizers”, J. Appl. Math. Comput., vol. 22, no. 1-2, pp. 403-410,...
  • S. J. Baishya, “On finite groups with specific number of centralizers”, Int. Electron. J. Algebra, vol. 13, pp. 53-62, 2013.
  • F. Barry, D. MacHale and A. N. Shé, “Some supersolvability conditions for finite groups”, Math. Proc. R. Ir. Acad., vol. 106, no. A2, pp....
  • S. M. Belcastro and G. J. Sherman, “Counting centralizers in finite groups”, Math. Magazine, vol. 67, no. 5, pp. 366-374, 1994.
  • S. M. Buckley and D. Machale, “Contrasting the commuting probabilities of groups and rings”. [On line]. Available: https://bit.ly/3sxa8b1
  • S. M. Buckley, D. Machale and A. N. Shé, “Finite rings with many commuting pairs of elements”. [On line]. Available: https://bit.ly/3QMAWfe
  • A. K. Das and R. K. Nath, “A characterization of certain finite groups of odd order”, Math. Proc. Royal Irish Acad., vol. 111, no. A2, pp....
  • A. K. Das, R. K. Nath and M. R. Pournaki, “A survey on the estimation of commutativity in finite groups”, Southeast Asian Bull. Math., vol....
  • J. Dutta and D. K. Basnet, “Some bounds for commuting probability of finite rings”, Proc. Math. Sci., vol. 129, no. 1, pp. 1-6, 2019.
  • J. Dutta, D. K. Basnet and R. K. Nath, “Characterizing some rings of finite order”, Tamkang J. Math., vol. 53, no. 2, pp. 97-108, 2022.
  • J. Dutta, D. K. Basnet and R. K. Nath, “A note on n-centralizer finite rings”, An. Stiint. Univ. Al. I. Cuza Iasi Mat., vol. 64, no. 1, pp....
  • J. Dutta, D. K. Basnet and R. K. Nath, “On commuting probability of finite rings,” Indag. Math., vol. 28, no. 2, pp. 372-382, 2017.
  • P. Dutta and R. K. Nath, “A generalization of commuting probability of finite rings”, Asian-European J. Math., vol. 11, Art. ID. 1850023,...
  • P. Dutta and R. K. Nath, “On relative commuting probability of finite rings”, Miskolc Math. Notes, vol. 20, no. 1, pp. 225-232, 2019.
  • P. Dutta and R. K. Nath, “On r-commuting probability of finite rings”, Indian Journal of Mathematics, vol. 62, no. 3, pp. 287-297, 2020.
  • P. Erdös and P. Turán, “On some problems of a statistical group-theory IV”, Acta. Math. Acad. Sci. Hungar., vol. 19, pp. 413-435, 1968.
  • A. Erfanian, R. Rezaei and P. Lescot, “On the relative commutativity of a subgroup of a finite group”, Comm. Algebra, vol. 35, pp. 4183-4197,...
  • I. V. Erovenko and B. Sury, “Commutativity degrees of wreath products of finite abelian groups”, Bull. Aust. Math. Soc., vol. 77, no. 1, pp....
  • R. M. Guralnick and G. R. Robinson, “On the commuting probability in finite groups”, J. Algebra, vol. 300, no. 2, pp. 509-528, 2006.
  • W. H. Gustafson, “What is the probability that two group elements commute?”, Amer. Math. Monthly, vol. 80, pp. 1031-1034, 1973.
  • P. Hegarty, “Limit points in the range of the commuting probability function on finite groups”, J. Group Theory, vol. 16, no. 2, pp. 235-247,...
  • K. H. Hofmann and F. G. Russo, “The probability that x and y commute in a compact group”, Math. Proc. Cambridge Phil. Soc., vol. 153, pp....
  • K. S. Joseph, Commutativity in Non-Abelian Groups, Ph.D. thesis, University of California, Los Angeles, 1969.
  • P. Lescot, “Isoclinism classes and commutativity degrees of finite groups”, J. Algebra, vol. 177, pp. 847-869, 1995.
  • P. Lescot, “Central extensions and commutativity degree”, Comm. Algebra, vol. 29, no. 10, pp. 4451-4460, 2001.
  • D. MacHale, ”Commutativity in finite rings”, Amer. Math. Monthly, vol. 83, pp. 30-32, 1976.
  • R. K. Nath, “Commutativity degree of a class of finite groups and consequences”, Bull. Aus. Math. Soc., vol. 88, no. 3, pp. 448-452, 2013.
  • R. K. Nath and A. K. Das, “On a lower bound of commutativity degree”, Rend. Circ. Mat. Palermo, vol. 59, pp. 137-142, 2010.
  • D. J. Rusin, “What is the probability that two elements of a finite group commute?”, Pacific J. Math., vol. 82, pp. 237-247, 1979.