Commuting probability for subrings and quotient rings

Authors

  • Stephen M. Buckley Department of Mathematics and Statistics, Maynooth University, Maynooth, Co. Kildare, Ireland ROR
  • Desmond MacHale School of Mathematical Sciences, University College Cork, Cork, Ireland ROR

Keywords:

Commuting probability, Subring, Quotient ring

Abstract

We prove that the commuting probability of a finite ring is no larger than the commuting probabilities of its subrings and quotients, and characterize when equality occurs in such a comparison.

Downloads

Download data is not yet available.

References

S. M. Buckley, Distributive algebras, isoclinism, and invariant probabilities, in: Groups, Rings, Group Rings, and Hopf Algebras, Contemporary Mathematics, vol. 634, American Mathematical Society, Providence, RI, 2015, pp. 19–34.

S. M. Buckley and D. MacHale, Commuting probability for subrings and quotient rings, J. Algebra Comb. Discrete Appl. 4(2) (2017) 163–173.

S. M. Buckley, D. MacHale, and Á. Ní Shé, Finite rings with many commuting pairs, Math. Proc. R. Ir. Acad. 114A(2) (2014) 157–170.

J. D. Dixon, Probabilistic group theory, C. R. Math. Acad. Sci. Soc. R. Can. 24(1) (2002) 1–15.

P. Erdős and P. Turán, On some problems of a statistical group–theory, IV, Acta Math. Acad. Sci. Hungar. 19(3–4) (1968) 413–435.

R. M. Guralnick and G. R. Robinson, On the commuting probability in finite groups, J. Algebra 300(2) (2006) 509–528.

K. S. Joseph, Commutativity in Non–Abelian Groups, Ph.D. thesis, University of California, Los Angeles, 1977.

D. MacHale, How commutative can a non–commutative group be?, Math. Gaz. 58(405) (1974) 199–202.

D. MacHale, Commutativity in finite rings, Amer. Math. Monthly 83(1) (1976) 30–32.

D. Rusin, What is the probability that two elements of a finite group commute?, Pacific J. Math. 82(1) (1979) 237–247.

Downloads

Published

2017-05-15

How to Cite

Buckley, S. M., & MacHale, D. (2017). Commuting probability for subrings and quotient rings. Journal of Algebra Combinatorics Discrete Structures and Applications, 4(2), 189–196. Retrieved from https://www.jacodesmath.com/index.php/jacodesmath/article/view/68

Issue

Section

Articles