Geodesic
Some Commutativity Theorems for Near-Rings with Left Multipliers
Kragujevac Journal of Mathematics, Tome 44 (2020) no. 2, p. 205 .

Voir la notice de l'article provenant de la source eLibrary of Mathematical Institute of the Serbian Academy of Sciences and Arts

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 $x, y\in U$.
Classification : 16Y30, 16N60, 16W25
Keywords: $3$-Prime near-ring, derivations, commutativity, generalized derivation, left multiplier 
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     author = {A. Boua and A. Y. Abdelwanis and A. Chillali},
     title = {Some {Commutativity} {Theorems} for {Near-Rings} with {Left} {Multipliers}},
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A. Boua; A. Y. Abdelwanis; A. Chillali. Some Commutativity Theorems for Near-Rings with Left Multipliers. Kragujevac Journal of Mathematics, Tome 44 (2020) no. 2, p. 205 . http://geodesic.mathdoc.fr/item/KJM_2020_44_2_a3/