IJPAM: Volume 83, No. 2 (2013)

PRIME GAMMA NEAR-RINGS WITH DERIVATIONS

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
MALAYSIA
$^{2,3}$Department of Mathematics
Rajshahi University
Rajshahi, 6205, BANGLADESH


Abstract. Let $N$ be a prime $\Gamma $near-ring with the center $Z(N)$. The objective of this paper is to study derivations on $N$. We prove two results:

(a) Let $N$ be 2-torsion free and let $D_{\mathrm{1}}$ and $D_{\mathrm{2}}$ be derivations on $N$ such that $D_{\mathrm{1}}D_{\mathrm{2}}$ is also a derivation. Then $D_{\mathrm{1}} = 0$ or $D_{\mathrm{2}} = 0$ if and only if $[D_{\mathrm{1}}(x), D_{\mathrm{2}}(y)]_{\mathrm{\alpha }} = 0$ for all $x$, $y\in N$, $\alpha \in \Gamma $;

(b) Let $n$ be an integer greater than 1, $N$ be $n$!-torsion free, and $D$ be a derivation with $D^{n}(N) = \{ 0\}$. Then $D(Z(N))= \{ 0\}$.

Received: June 13, 2012

AMS Subject Classification: 16W25, 16Y30, 16Y99

Key Words and Phrases: commutative ring, non commutative ring, derivation

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DOI: 10.12732/ijpam.v83i2.1 How to cite this paper?
Source:
International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2013
Volume: 83
Issue: 2