Commuting probability for subrings and quotient rings

Öz

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.

Anahtar Kelimeler

• Commuting probability,Subring,Quotient ring

Kaynakça

1. [1] S. M. Buckley, Distributive algebras, isoclinism, and invariant probabilities, Contemp. Math. 634 (2015) 31–52.
2. [2] S. M. Buckley, D. MacHale, Commuting probabilities of groups and rings, preprint.
3. [3] S. M. Buckley, D. MacHale, Á. Ní Shé, Finite rings with many commuting pairs of elements, preprint.
4. [4] J. D. Dixon, Probabilistic group theory, C. R. Math. Acad. Sci. Soc. R. Can. 24(1) (2002) 1–15.
5. [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. [6] R. M. Guralnick, G. R. Robinson, On the commuting probability in finite groups, J. Algebra 300(2) (2006) 509–528.
7. [7] K. S. Joseph, Commutativity in non–abelian groups, PhD thesis, University of California, Los Angeles, 1969.
8. [8] D. MacHale, How commutative can a non–commutative group be? Math. Gaz. 58(405) (1974) 199–202.

9. [9] D. MacHale, Commutativity in finite rings, Amer. Math. Monthly 83(1) (1976) 30–32.
10. [10] D. Rusin, What is the probability that two elements of a finite group commute?, Pacific J. Math. 82(1) (1979) 237–247.