COMMUTATIVITY AND DECOMPOSITION FOR NEAR RINGS

Let $R$ be a distributively generated (d.g) near ring satisfy one of the following condit ions. (*) For each $x$, $y$ in $R$, there exists a positive integer $n =n(x,y)$ such that $xy =(yx)^n$.(**) For each $x$, $y$ in $R$, there exist positive integers $m = m(x,y)$ and $n = n(x,y)$ for which $xy = y^m x^n$. In [2], Bell proved the commutativity of $R$ satisfying (*) or (**) under appropriate additional hypothesis. In this paper, we generalize the above properties for wider class of near rings known as D-near rings. Also we provide an example for justification of our results. Furthermore, we give a decomposition Theorem for near rings satisfying (**).