#### 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 $a_{\gamma1}x_{\gamma2}a\beta1y\beta2a$ = $a\beta1y\beta2a_{\gamma1}x_{\gamma2}a$ for all x,y $\in$ M and $\gamma1,\gamma2,\beta1,\beta2\in\Gamma$. 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 x m over a division ring D and $\Gamma$ is a nonzero subgroup of the additive abelian group of all rectangular matrices of type m x n over a division ring D. Furthermore M is the $\Gamma$-ring of all n x 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 $q\gamma1x\gamma2\beta1y\beta2q$ =$q\beta1y\beta2q\gamma1x\gamma2q$ for all x $in$ M and $\beta,\gamma\in\Gamma$, then a and b are $C_{\Gamma}$-dependent. (3) Let M be prime $\Gamma$-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 $q\gamma1x\gamma2q\beta1y\beta2q$ = $q\beta1y\beta2q\gamma1x\gamma2q$ for all x,y $\in$ M, $\gamma1,\gamma2,\beta1,\beta2\in\Gamma$ 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.

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 $a_{\gamma1}x_{\gamma2}a\beta1y\beta2a$ = $a\beta1y\beta2a_{\gamma1}x_{\gamma2}a$ for all x,y $\in$ M and $\gamma1,\gamma2,\beta1,\beta2\in\Gamma$. 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 x m over a division ring D and $\Gamma$ is a nonzero subgroup of the additive abelian group of all rectangular matrices of type m x n over a division ring D. Furthermore M is the $\Gamma$-ring of all n x 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 $q\gamma1x\gamma2\beta1y\beta2q$ =$q\beta1y\beta2q\gamma1x\gamma2q$ for all x $in$ M and $\beta,\gamma\in\Gamma$, then a and b are $C_{\Gamma}$-dependent. (3) Let M be prime $\Gamma$-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 $q\gamma1x\gamma2q\beta1y\beta2q$ = $q\beta1y\beta2q\gamma1x\gamma2q$ for all x,y $\in$ M, $\gamma1,\gamma2,\beta1,\beta2\in\Gamma$ 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.