Regularity of the generalized centroid of semi-prime gamma rings

Commun. Korean Math. Soc. 2004 Vol. 19, No. 2, 233-242 Printed June 1, 2004

Mehmet Ali Ozturk, Young Bae Jun Cumhuriyet University, Gyeongsang National University

Abstract : 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.