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.
Keywords : semi-prime $\Gamma$-ring, generalized centroid, (regular, prime, quotient) $\Gamma$-ring
MSC numbers : 16N60, 16Y30, 16A76, 16Y99
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