Turkish Journal of Mathematics
DOI

Abstract
The aim of this paper is to study the properities of the extended centroid of the prime \Gammarings. Main results are the following theorems: (1) Let M be a simple \Gammaring 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 \Gammaring 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 \Gammaring of all n\times n matrices over the field C_{\Gamma}. (2) Let M be a prime \Gammaring and C_{\Gamma} the extended centroid of M. If a and b are nonzero 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 \GammaFring, Q quotient \Gammaring of M and C_{\Gamma} the extended centroid of M. If q is nonzero 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 \Gammaring 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
\Gammadivision ring, \Gammafield, extented centroid, central closure.
First Page
367
Last Page
377
Recommended Citation
ÖZTÜRK, MEHMET ALİ and JUN, YOUNG BAE (2001) "On the Centroid of the Prime Gamma Rings II," Turkish Journal of Mathematics: Vol. 25: No. 3, Article 2. Available at: https://journals.tubitak.gov.tr/math/vol25/iss3/2