ON A COMBINATORIAL COMMUTATIVITY CONDITION FOR RINGS

Let k >= 2. A ring R is called a B-k-ring if vertical bar K-2 vertical bar <= (k+1 2) 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.