TY - JOUR ID - 3080 TI - Identities in $3$-prime near-rings with left multipliers JO - Journal of Algebra and Related Topics JA - JART LA - en SN - 2345-3931 AU - Ashraf, M. AU - Boua, A. AD - Department of Mathematics, Faculty of Science, Aligarh Muslim University, Aligarh 202002, India AD - Department of Mathematics, Physics and Computer Science, Sidi Mohammed Ben Abdellah University,Taza, Morocco Y1 - 2018 PY - 2018 VL - 6 IS - 1 SP - 67 EP - 77 KW - $3$-Prime near-ring KW - derivations KW - commutativity KW - left multiplier DO - 10.22124/jart.2018.10093.1096 N2 - Let $\mathcal{N}$ be a $3$-prime near-ring with the center$Z(\mathcal{N})$ and $n \geq 1$ be a fixed positive integer. Inthe present paper it is shown that a $3$-prime near-ring$\mathcal{N}$ is a commutative ring if and only if it admits aleft multiplier $\mathcal{F}$ satisfying any one of the followingproperties: $(i)\:\mathcal{F}^{n}([x, y])\in Z(\mathcal{N})$, $(ii)\:\mathcal{F}^{n}(x\circ y)\in Z(\mathcal{N})$,$(iii)\:\mathcal{F}^{n}([x, y])\pm(x\circ y)\in Z(\mathcal{N})$ and $(iv)\:\mathcal{F}^{n}([x, y])\pm x\circ y\in Z(\mathcal{N})$, for all $x, y\in\mathcal{N}$. UR - https://jart.guilan.ac.ir/article_3080.html L1 - https://jart.guilan.ac.ir/article_3080_2bfb17a33939beaf9b8352a63a5aa703.pdf ER -