Identities in $3$-prime near-rings with left multipliers

Document Type : Research Paper


1 Department of Mathematics, Faculty of Science, Aligarh Muslim University, Aligarh 202002, India

2 Department of Mathematics, Physics and Computer Science, Sidi Mohammed Ben Abdellah University,Taza, Morocco


Let $\mathcal{N}$ be a $3$-prime near-ring with the center
$Z(\mathcal{N})$ and $n \geq 1$ be a fixed positive integer. 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 a
left multiplier $\mathcal{F}$ satisfying any one of the following
properties: $(i)\:\mathcal{F}^{n}([x, y])\in Z(\mathcal{N})$, $(ii)\:\mathcal{F}^{n}(x\circ y)\in Z(\mathcal{N})$,
$(iii)\:\mathcal{F}^{n}([x, y])\pm(x\circ y)\in Z(\mathcal{N})$ and $(iv)\:\mathcal{F}^{n}([x, y])\pm x\circ y\in Z(\mathcal{N})$, for all $x, y\in\mathcal{N}$.