WEAKLY NOVIKOV RINGS WITH ASSOCIATORS IN THE NUCLEI

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. In this paper we discuss some results on the rings with the generalized 
commutators in the nuclei. Here we assume R to satisfy weakly Novikov identity (w, x, yz) = y(w, x, z), for all    
w,x,y,z in R. We prove that if R is a prime Weakly Novikov ring such that commutator contained in two of the 
three nuclei, then R is associative or commutative. Also we prove that if R is a semi prime weakly M-ring i.e., 
(w, xy, z) = x(w, y, z), for all w,x,y,z in R. with (R, R, R) ⊆ Nm, then R is associative.