# Commuting probability for subrings and quotient rings

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.

### References

S. M. Buckley, Distributive algebras, isoclinism, and invariant probabilities, Contemp. Math. 634 (2015) 31–52.

S. M. Buckley, D. MacHale, Commuting probabilities of groups and rings, preprint.

S. M. Buckley, D. MacHale, Á. Ní Shé, Finite rings with many commuting pairs of elements, preprint.

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

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

R. M. Guralnick, 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, PhD thesis, University of California, Los Angeles, 1969.

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.

S. M. Buckley, D. MacHale, Commuting probabilities of groups and rings, preprint.

S. M. Buckley, D. MacHale, Á. Ní Shé, Finite rings with many commuting pairs of elements, preprint.

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

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

R. M. Guralnick, 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, PhD thesis, University of California, Los Angeles, 1969.

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.

Published

2017-05-15

Section

Articles