IJPAM: Volume 102, No. 3 (2015)

$\Gamma^*$-DERIVATION ACTING AS
AN ENDOMORPHISM AND AS AN ANTI-ENDOMORPHISM
IN SEMIPRIME $\Gamma$-RING M WITH INVOLUTION

Ali Kareem Kadhim$^1$, Hajar Sulaiman$^2$, Abdul-Rahman Hammed Majeed$^3$
$^{1,2}$School of Mathematical Sciences
Universiti Sains Malaysia, 11800 USM
Penang, MALAYSIA
$^3$Department of Mathematics
University of Baghdad
Baghdad, Iraq


Abstract. Let M be a semiprime $\Gamma$-ring with involution satisfying the condition that $ a\alpha b\beta c = a\beta b\alpha c$ $(a,~b,~c \in M $ and $\alpha,\beta \in \Gamma)$. An additive mapping $d:M\rightarrow M$ is called $\Gamma^*$-derivation if $d(x\alpha y)=d(x)\alpha y^*+x\alpha d(y)$. In this paper we will prove that if d is $\Gamma^*$-derivation of a semiprime $\Gamma$-ring with involution which is either an endomorphism or anti-endomorphism, then d=0.

Received: March 25, 2015

AMS Subject Classification: 16W10, 16W25, 16N60

Key Words and Phrases: endomorphism of $\Gamma$-ring M, semiprime $\Gamma$-ring with involution, $\Gamma^*$-derivation

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DOI: 10.12732/ijpam.v102i3.7 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: 2015
Volume: 102
Issue: 3
Pages: 495 - 501


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