IJPAM: Volume 82, No. 5 (2013)


Isamiddin S. Rakhimov$^1$, Kalyan Kumar Dey$^2$, Akhil Chandra Paul$^3$
$^1$Department of Mathematics, FS
Institute for Mathematical Research (INSPEM)
Universiti Putra Malaysia
$^{2,3}$Department of Mathematics
Rajshahi University
Rajshahi-6205, BANGLADESH

Abstract. Let $N$ be a 2 torsion free prime $\Gamma $-near-ring with center $Z(N)$ and let $d$ be a nontrivial derivation on $N$ such that $d(N) \subseteq Z(N)$. Then we prove that $N$ is commutative. Also we prove that if $d$ be a nonzero ($\sigma $,$\tau
)$-derivation on $N$ such that $d(N)$ commutes with an element $a $of$N$ then ether $d$ is trivial or $a $ is in $Z(N)$. Finally if $d_{{1}}$ be a nonzero ($\sigma $,$\tau
)$-derivation and $d_{{2}}$ be a nonzero derivation on $N$ such that $d_{{1}}\tau = \tau d_{{1}}$, $d_{{1}}\sigma = \sigma d_{{1}}$, $d_{{2}}\tau = \tau d_{{2}}$, $d_{{2}}\sigma $ $= \sigma d_{{2\thinspace }}$with $d_{{1}}(N)\Gamma
\sigma (d_{{2}}(N))$ $= \tau (d_{{2}}(N))$ $\Gamma
d_{{1}}(N)$ then $N$ is a commutative $\Gamma $-ring.

Received: May 24, 2012

AMS Subject Classification: 16Y30, 19W25, 16U80

Key Words and Phrases: ring, prime ring, derivation

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DOI: 10.12732/ijpam.v82i5.1 How to cite this paper?
International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2013
Volume: 82
Issue: 5