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Commuting probability for subrings and quotient rings

Year 2017, Volume: 4 Issue: 2 (Special Issue: Noncommutative rings and their applications), 189 - 196, 10.01.2017
https://doi.org/10.13069/jacodesmath.284962

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.

References

  • [1] S. M. Buckley, Distributive algebras, isoclinism, and invariant probabilities, Contemp. Math. 634 (2015) 31–52.
  • [2] S. M. Buckley, D. MacHale, Commuting probabilities of groups and rings, preprint.
  • [3] S. M. Buckley, D. MacHale, Á. Ní Shé, Finite rings with many commuting pairs of elements, preprint.
  • [4] J. D. Dixon, Probabilistic group theory, C. R. Math. Acad. Sci. Soc. R. Can. 24(1) (2002) 1–15.
  • [5] P. Erdös, P. Turán, On some problems of a statistical group–theory, IV, Acta Math. Acad. Sci. Hung. 19(3) (1968) 413–435.
  • [6] R. M. Guralnick, G. R. Robinson, On the commuting probability in finite groups, J. Algebra 300(2) (2006) 509–528.
  • [7] K. S. Joseph, Commutativity in non–abelian groups, PhD thesis, University of California, Los Angeles, 1969.
  • [8] D. MacHale, How commutative can a non–commutative group be? Math. Gaz. 58(405) (1974) 199–202.
  • [9] D. MacHale, Commutativity in finite rings, Amer. Math. Monthly 83(1) (1976) 30–32.
  • [10] D. Rusin, What is the probability that two elements of a finite group commute?, Pacific J. Math. 82(1) (1979) 237–247.
Year 2017, Volume: 4 Issue: 2 (Special Issue: Noncommutative rings and their applications), 189 - 196, 10.01.2017
https://doi.org/10.13069/jacodesmath.284962

Abstract

References

  • [1] S. M. Buckley, Distributive algebras, isoclinism, and invariant probabilities, Contemp. Math. 634 (2015) 31–52.
  • [2] S. M. Buckley, D. MacHale, Commuting probabilities of groups and rings, preprint.
  • [3] S. M. Buckley, D. MacHale, Á. Ní Shé, Finite rings with many commuting pairs of elements, preprint.
  • [4] J. D. Dixon, Probabilistic group theory, C. R. Math. Acad. Sci. Soc. R. Can. 24(1) (2002) 1–15.
  • [5] P. Erdös, P. Turán, On some problems of a statistical group–theory, IV, Acta Math. Acad. Sci. Hung. 19(3) (1968) 413–435.
  • [6] R. M. Guralnick, G. R. Robinson, On the commuting probability in finite groups, J. Algebra 300(2) (2006) 509–528.
  • [7] K. S. Joseph, Commutativity in non–abelian groups, PhD thesis, University of California, Los Angeles, 1969.
  • [8] D. MacHale, How commutative can a non–commutative group be? Math. Gaz. 58(405) (1974) 199–202.
  • [9] D. MacHale, Commutativity in finite rings, Amer. Math. Monthly 83(1) (1976) 30–32.
  • [10] D. Rusin, What is the probability that two elements of a finite group commute?, Pacific J. Math. 82(1) (1979) 237–247.
There are 10 citations in total.

Details

Subjects Engineering
Journal Section Articles
Authors

Stephen M. Buckley

Desmond Machale

Publication Date January 10, 2017
Published in Issue Year 2017 Volume: 4 Issue: 2 (Special Issue: Noncommutative rings and their applications)

Cite

APA Buckley, S. M., & Machale, D. (2017). Commuting probability for subrings and quotient rings. Journal of Algebra Combinatorics Discrete Structures and Applications, 4(2 (Special Issue: Noncommutative rings and their applications), 189-196. https://doi.org/10.13069/jacodesmath.284962
AMA Buckley SM, Machale D. Commuting probability for subrings and quotient rings. Journal of Algebra Combinatorics Discrete Structures and Applications. May 2017;4(2 (Special Issue: Noncommutative rings and their applications):189-196. doi:10.13069/jacodesmath.284962
Chicago Buckley, Stephen M., and Desmond Machale. “Commuting Probability for Subrings and Quotient Rings”. Journal of Algebra Combinatorics Discrete Structures and Applications 4, no. 2 (Special Issue: Noncommutative rings and their applications) (May 2017): 189-96. https://doi.org/10.13069/jacodesmath.284962.
EndNote Buckley SM, Machale D (May 1, 2017) Commuting probability for subrings and quotient rings. Journal of Algebra Combinatorics Discrete Structures and Applications 4 2 (Special Issue: Noncommutative rings and their applications) 189–196.
IEEE S. M. Buckley and D. Machale, “Commuting probability for subrings and quotient rings”, Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 4, no. 2 (Special Issue: Noncommutative rings and their applications), pp. 189–196, 2017, doi: 10.13069/jacodesmath.284962.
ISNAD Buckley, Stephen M. - Machale, Desmond. “Commuting Probability for Subrings and Quotient Rings”. Journal of Algebra Combinatorics Discrete Structures and Applications 4/2 (Special Issue: Noncommutative rings and their applications) (May 2017), 189-196. https://doi.org/10.13069/jacodesmath.284962.
JAMA Buckley SM, Machale D. Commuting probability for subrings and quotient rings. Journal of Algebra Combinatorics Discrete Structures and Applications. 2017;4:189–196.
MLA Buckley, Stephen M. and Desmond Machale. “Commuting Probability for Subrings and Quotient Rings”. Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 4, no. 2 (Special Issue: Noncommutative rings and their applications), 2017, pp. 189-96, doi:10.13069/jacodesmath.284962.
Vancouver Buckley SM, Machale D. Commuting probability for subrings and quotient rings. Journal of Algebra Combinatorics Discrete Structures and Applications. 2017;4(2 (Special Issue: Noncommutative rings and their applications):189-96.