چکیده انگلیسی:
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: $(i):mathcal{F}^{n}([x, y])in Z(mathcal{N})$, $(ii):mathcal{F}^{n}(xcirc y)in Z(mathcal{N})$, $(iii):mathcal{F}^{n}([x, y])pm(xcirc y)in Z(mathcal{N})$ and $(iv):mathcal{F}^{n}([x, y])pm xcirc yin Z(mathcal{N})$, for all $x, yinmathcal{N}$.
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