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/