On the centroid of the prime gamma rings II

Abstract

The aim of this paper is to study the properties of the extended centroid of the prime ?-rings. Main results are the following theorems: (1) Let M be a simple ?-ring with unity. Suppose that for some a ? 0 in M we have a?1x?2a?1y?2a = a?1y?2a?1x?2a for all x,y ? M and ?1,?2, ?1,?2 ? ?. Then M is isomorphic onto the ?-ring Dn,m, where Dn,m is the additive abelian group of all rectangular matrices of type n × m over a division ring D and ? is a nonzero subgroup of the additive abelian group of all rectangular matrices of type m × n over a division ring D. Furthermore M is the ?-ring of all n × n matrices over the field C?. (2) Let M be a prime ?-ring and C? the extended centroid of M. If a and b are non-zero elements in S = M?C? such that a?x?b = b?x?a for all x ? M and ?, ? ? ?, then a and b are C?-dependent. (3) Let M be prime ?-ring, Q quotient ?-ring of M and C? the extended centroid of M. If q is non-zero element in Q such that q?1x?2q?1y?2q = q?1y?2q?1x? 2q for all x, y ? M, ?1, ?2, ?1, ?2 ? ? then S is a primitive ?-ring with minimal right ( left ) ideal such that e?S, where e is idempotent and C??e is the commuting ring of S on e?S. © TÜBİTAK.