An Orthogonal Left Centralizer and Reverse Left Centralizer on Semiprime -Rings

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Fawaz Ra'ad Jarullah , Yilmaz Çeven

Abstract

Let M be a semiprime G-ring . In this paper we introduce the concept of orthogonal left centralizer and reverse left centralizer on a semiprime G- ring and we prove the following       main result: Let M be a 2-torsion free semiprime G- ring, t be a left centralizer and h be a reverse left centralizer of M , such that  xazby = xbzay , for all  x , y , z Î M , a , b Î G and t , h are commuting. Then t and h are orthogonal if and only if     t(x) G M G h(y) + h(x) G M G t(y) = (0) , for all x , y Î M .

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