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# On the Centroid of the Prime Gamma Rings II

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## Abstract

The aim of this paper is to study the properities of the extended centroid of the prime \Gamma-rings. Main results are the following theorems: (1) Let M be a simple \Gamma-ring with unity. Suppose that for some a\neq 0 in M we have a\gamma_{1} x\gamma_{2} a\beta_{1} y\beta_{2}a = a\beta_{1} y\beta_{2} a\gamma_{1} x\gamma_{2}a for all x, y\in M and \gamma_{1} ,\gamma_{2} ,\beta_{1} ,\beta_{2} \in \Gamma. Then M is isomorphic onto the \Gamma-ring D_{n,m}, where D_{n,m} is the additive abelian group of all rectangular matrices of type n\times m over a division ring D and \Gamma is a nonzero subgroup of the additive abelian group of all rectangular matrices of type m\times n over a division ring D. Furthermore M is the \Gamma-ring of all n\times n matrices over the field C_{\Gamma}. (2) Let M be a prime \Gamma-ring and C_{\Gamma} the extended centroid of M. If a and b are non-zero elements in S=M\Gamma C_{\Gamma} such that a\gamma x\beta b = b\beta x\gamma a for all x \in M and \beta ,\gamma \in \Gamma, then a and b are C_{\Gamma}-dependent. (3) Let M be prime \GammaF-ring, Q quotient \Gamma-ring of M and C_{\Gamma} the extended centroid of M. If q is non-zero element in Q such that q\gamma_{1} x\gamma_{2}q\beta_{1}y\beta_{2}q = q\beta_{1}y\beta_{2}q\gamma_{1}x\gamma_{2}q for all x, y\in M, \gamma_{1} , \gamma_{2}, \beta_{1}, \beta _{2} \in \Gamma then S is a primitive \Gamma-ring with minimal right ( left ) ideal such that e\Gamma S, where e is idempotent and C_{\Gamma}\Gamma e is the commuting ring of S on e\Gamma S.

## Keywords

\Gamma-division ring, \Gamma-field, extented centroid, central closure.

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