Geodesic
On Gamma-Rings with $(\sigma,\tau)$-Skew-Commuting and $(\sigma,\tau)$-Skew-Centralizing Mappings
Kragujevac Journal of Mathematics, Tome 42 (2018) no. 1, p. 41

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

Let $M$ be a 2-torsion free $\Gamma$-ring with left identity $e$. Let $D : M x M\rightarrow M$ be a symmetric bi-additive mapping and $d(x) = D(x, x)$. Let $\sigma$ and $\tau$ be an endomorphism and an epimorphism of $M$, respectively. We prove the following: \begin{itemize} em[(i)] if $d$ is $(\sigma ,\tau )$-skew-commuting on $M$, then $D = 0$; em[(ii)] if $d$ is $(\tau ,\tau )$-skew-centralizing on $M$, then $d$ is $(\tau ,\tau )$-commuting on $M$; em[(iii)] if $M$ is a 3-torsion free $\Gamma$-ring satisfying $x\alpha y\beta z=x \beta y \alpha z$ for all $x, y, z\in M$ and $\alpha , \beta \in \Gamma$, then 2-$(\sigma , \tau )$-commutingness of $d$ on $M$ implies its $(\sigma ,\tau )$-commutingness. \end{itemize}
Classification : 16W20 16Y99
Keywords: $\Gamma$-ring, $(\sigma ;\tau )$-skew-commuting mappings, $(\sigma;\tau)$-skew-centralizing mappings, $(\sigma;\tau)$-commuting mappings 
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     author = {Kalyan Kumar Dey and Akhil Chandra Paul and Bijan Davvaz},
     title = {On {Gamma-Rings} with $(\sigma,\tau)${-Skew-Commuting} and $(\sigma,\tau)${-Skew-Centralizing} {Mappings}},
     journal = {Kragujevac Journal of Mathematics},
     pages = {41 },
     publisher = {mathdoc},
     volume = {42},
     number = {1},
     year = {2018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KJM_2018_42_1_a3/}
}
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Kalyan Kumar Dey; Akhil Chandra Paul; Bijan Davvaz. On Gamma-Rings with $(\sigma,\tau)$-Skew-Commuting and $(\sigma,\tau)$-Skew-Centralizing Mappings. Kragujevac Journal of Mathematics, Tome 42 (2018) no. 1, p. 41 . http://geodesic.mathdoc.fr/item/KJM_2018_42_1_a3/