Characterizing Some Rings of Finite Order

Article Sidebar

PDF
Published: May 1, 2022
DOI: https://doi.org/10.5556/j.tkjm.53.2022.3370
Keywords:
Finite ring centralizer commutativity degree

Main Article Content

Rajat Kanti Nath
https://orcid.org/0000-0003-4766-6523
Jutirekha Dutta
Tezpur University
Dhiren Basnet
Tezpur University
https://orcid.org/0000-0002-9589-9287

Abstract




In this paper, we compute the number of distinct centralizers of some classes of finite rings. We then characterize all finite rings with $n$ distinct centralizers for any positive integer $n \le 5$. Further we give some connections between the number of distinct centralizers of a finite ring and its commutativity degree.




Article Details

How to Cite
Nath, R. K., Dutta, J., & Basnet, D. (2022). Characterizing Some Rings of Finite Order. Tamkang Journal of Mathematics, 53(2), 97–108. https://doi.org/10.5556/j.tkjm.53.2022.3370
Issue
Vol. 53 No. 2 (2022)
Section
Papers
Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

References

A. R. Ashrafi, On finite groups with a given number of centralizers, Algebra Colloq., 7 (2000), 139–146.

M. Behboodi, R. Beyranvand, A.Hashemi and H. Khabazian, Classification of finite rings : theory and algorithm, Czechoslovak Math. J., 64 (2014), 641–658.

S. M. Belcastro and G. J. Sherman, Counting centralizers in finite groups, Math. Magazine, 67 (1994), 366–374.

M. Bruckheimer, A. C. Bryan and A. Muir, Groups which are the union of three sub- groups, Amer. Math. Monthly, 77 (1970), 52–57.

S. M. Buckley, D. MacHale and A. NiShe, Finite rings with many commuting pairs of elements. Available from:

http://archive.maths.nuim.ie/staff/sbuckley/Papers/bms.pdf.

S. M. Buckley and D. MacHale, Contrasting the commuting probabilities of groups and rings. Available from:

http://archive.maths.nuim.ie/staff/sbuckley/Papers/bm_g-vs-r.pdf.

J. H. E. Cohn, On $n$-sum groups, Math. Scand., 75(1994), 44–58.

C. J. Chikunji, A Classification of a certain class of completely primary finite rings, Ring and Module Theory, Trends in Mathematics 2010, Springer Basel, pp 83–90.

J. Dutta, D. K. Basnet and R. K. Nath, A note on $n$-centralizer finite rings, An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.), LXIV (2018), 161–171.

J. B. Derr, G. F. Orr and P. S. Peck, Noncommutative rings of order p4, J. Pure Appl. Algebra, 97 (1994), 109–116.

K. E. Eldridge, Orders for finite noncommutative rings with unity, Amer. Math. Monthly, 75 (1968), 512–514.

B. Fine, Classification of finite rings of order $p^2$, Math. Magazine, 66(1993), 248–252.

R. W. Goldbach and H. L. Claasen, Classification of not commutative rings with identity of order dividing $p^4$, Indag. Math., 6 (1995), 167–187.

D. MacHale, Commutativity in finite rings, Amer. Math. Monthly, 83(1976), 30–32.

R. Raghavendran, A class of finite rings, Compositio Math., 22(1970), 49–57.