RINGS WITH ASSOCIATORS IN THE RIGHT NUCLEUS

Abstract

The associator (x, y, z) is defined by (x, y, z) = (xy) z – x (yz) for all x, y, z in a ring. The commutator 
(x, y) is defined by (x, y) = xy – yx for all x, y in a ring. This can be considered to be a measure of non
commutativity of ring. By the nucleus N of a ring R, we mean the set of all elements n in R such that (n, R, R) = 
(R, n, R) = (R, R, n) = 0. A ring R is said to be of characteristic ≠ n if n x = 0, impels x = 0 for all x∈R and n is a 
natural number. A ring R is of characteristic ≠n is simply denoted by char. ≠n. In this paper we prove that if 
a ring R is simple or prime or semiprime of char. ≠2 with the associators in the right nucleus, then R is 
associative.