EXPERIMENTING WITH THE IDENTITY (XY) Z = Y (ZX)

Abstract

A non-empty set G together with a binary operation is called a quasi group or groupoid.We knows 
that a groupoid is a nonempty set with a single binary operation. For a positive integer k, we say that a 
groupoid is k-nice if the product of any k elements is the same, regardless of their association or order. With 
this, commutativity is then equivalent to being 2-nice. A groupoid is commutative and associative if and only if 
it is both 2-nice and 3-nice. In this paper we show that groupoids satisfying identity (xy) z = y (zx) are k-nice 
for each k  ≥ 5. Also we see that this yields the corollary that any semiprime ring satisfying