Characterization and commuting probability of n-centralizer finite rings
DOI:
https://doi.org/10.22199/issn.0717-6279-5440Keywords:
finite ring, commuting probability, n-centralizer ringsAbstract
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.
References
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, 2007.
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, 2006.
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. 163-177, 2006.
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. 69-78, 2011.
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. 37, pp. 161-180, 2013.
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. 161-171, 2018.
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, 2018.
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, 2007.
I. V. Erovenko and B. Sury, “Commutativity degrees of wreath products of finite abelian groups”, Bull. Aust. Math. Soc., vol. 77, no. 1, pp. 31-36, 2008.
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, 2013.
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. 557-571, 2012.
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.
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