Some Commutativity Theorems for Near-Rings with Left Multipliers

A. Boua, A. Y. Abdelwanis, A. Chillali

Let $\mathcal{N}$ be a $3$-prime near-ring with the center $Z(\mathcal{N})$, and $U$ be a nonzero semigroup ideal of $\mathcal{N}$. In the present paper it is shown that a $3$-prime near-ring $\mathcal{N}$ is a commutative ring if and only if it admits left multipliers $\mathcal{F}$ and $G$ satisfying any one of the following properties: ${\rm(i)}\:\mathcal{F}(x)G(y)±[x, y]\in Z(\mathcal{N})$; ${\rm(ii)}\:\mathcal{F}(x)G(y)±x\circ y\in Z(\mathcal{N})$; ${\rm(iii)}\:\mathcal{F}(x)G(y)±yx\in Z(\mathcal{N})$; ${\rm(iv)}\:\mathcal{F}(x)G(y)±xy\in Z(\mathcal{N})$ and ${\rm(v)}\:\mathcal{F}([x, y])±G(x\circ y)\in Z(\mathcal{N})$ for all $x, y\in U$.