STRONGLY RIGHT ALTERNATIVE RINGS AND BOL CIRCLE LOOPS”

Abstract

The associator (x, y, z) is defined by (x, y, z) = (xy) z - x (yz) for all x, y, z in R. This plays a central 
role in the study of non- associative rings. The commutator is defined by (x, y) = xy - yx for all x, y in R. A non
associative ring R is called right alternative if (x, y, y) = 0 for all x, y in R. A groupoid with an identity element is 
called a loop. A loop is (right) Bol if it satisfies the right Bol identity (xy.z) y = x (yz. y) and moufang if it 
satisfies the stronger identity (xy.z) y = x (y.zy).In this paper we present some results on strongly right 
alternative rings which become Bol loops under the circle operation. Also we see that the smallest Bol circle 
loop has order 16 and there are six such loops.