초록
The aim of this note is to study properties of the generalized centroid of the semi-prime gamma rings. Main results are the following theorems: (1) Let M be a semi-prime $\Gamma$-ring and Q a quotient $\Gamma$-ring of M. If W is a non-zero submodule of the right (left) M-module Q, then $W\Gamma$W $\neq 0. Furthermore Q is a semi-prime $\Gamma$-ring. (2) Let M be a semi-prime $\Gamma$-ring and $C_{{Gamma}$ the generalized centroid of M. Then $C_{\Gamma}$ is a regular $\Gamma$-ring. (3) Let M be a semi-prime $\Gamma$-ring and $C_{\gamma}$ the extended centroid of M. If $C_{\gamma}$ is a $\Gamma$-field, then the $\Gamma$-ring M is a prime $\Gamma$-ring.
