On a Combinatorial Commutativity Condition for Rings

Abstract

Let k ≥ 2. A ring R is called a B _k -ring if for all k-subsets K. All commutative rings are B _k -rings, hence it is natural to seek conditions which imply that a B _k -ring is commutative. We present some commutativity results for arbitrary k, and then focus on B ₄-rings and B ₅-rings with 1. One of our results asserts that, if R is a finite 2-torsion-free B ₄-ring or B ₅-ring with 1, then R is commutative.

Key Words:

• Commutativity
• B _k -groups
• B _k -rings

2000 Mathematics Subject Classification:

• Primary 16U80
• Secondary 16U60

ACKNOWLEDGMENT

We would like to thank the referee for useful suggestions which improved the readability of the paper.

This research was supported in part by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada.

Notes

Communicated by E. Puczylowski