Publication: Structure of certain near-rings
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Date
1992
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Università degli Studi di Trieste. Dipartimento di Scienze Matematiche
Abstract
Recentemente Ligh e Luh $\left[6\right]$ hanno trovato una scomposizione
in somma diretta per gli anelli che hanno la proprietà $\left(xy\right)^{n\left(x,y\right)}=xy$
usando la commutatività di questi anelli provata da Searcòid e MacHale
$\left[8\right]$. In questo lavoro si continua questo studio e si
ottiene una scomposizione per i quasi-anelli che hanno una qualunque
delle proprietà $\left(i\right)\left(xy\right)^{n\left(x,y\right)}=xy\left(ii\right)x^{n\left(x,y\right)}y^{m\left(xy\right)}=xy$
e $\left(iii\right)y^{m\left(xy\right)}x^{n\left(x,y\right)}=xy$
Using commutativity of rings satisfying $\left(xy\right)^{n\left(x,y\right)}=xy$ proved by Searcòid e MacHale $\left[8\right]$, recently Ligh and Luh $\left[6\right]$ have given-direct sum decomposition for rings with the mentioned condition. More recently Bell and Ligh $\left[3\right]$ sharpened this result and also established a structure of the near-rings satisfying $\left(xy\right)^{n\left(x,y\right)}=yx$. In the present paper we continue these investigations and obtain decomposition for near-rings satisfying any of the conditions $\left(i\right)\left(xy\right)^{n\left(x,y\right)}=xy\left(ii\right)x^{n\left(x,y\right)}y^{m\left(xy\right)}=xy$ and $\left(iii\right)y^{m\left(xy\right)}x^{n\left(x,y\right)}=xy$.
Using commutativity of rings satisfying $\left(xy\right)^{n\left(x,y\right)}=xy$ proved by Searcòid e MacHale $\left[8\right]$, recently Ligh and Luh $\left[6\right]$ have given-direct sum decomposition for rings with the mentioned condition. More recently Bell and Ligh $\left[3\right]$ sharpened this result and also established a structure of the near-rings satisfying $\left(xy\right)^{n\left(x,y\right)}=yx$. In the present paper we continue these investigations and obtain decomposition for near-rings satisfying any of the conditions $\left(i\right)\left(xy\right)^{n\left(x,y\right)}=xy\left(ii\right)x^{n\left(x,y\right)}y^{m\left(xy\right)}=xy$ and $\left(iii\right)y^{m\left(xy\right)}x^{n\left(x,y\right)}=xy$.
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Citation
Murtaza A. Quadri, Mohd. Ashraf, Asma Ali, “Structure of certain near-rings”, in: Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics, 24 (1992), pp. 161-167.