On Left Centralizers of Semiprime Γ-Rings
Keywords:Semiprime ?-ring, Left (right) centralizer, Centralizer, Commuting mapping, Centralizing mapping, Extended centroid.
Let M be a semiprime G-ring satisfying an assumption xaybz = xbyaz for all x, y, z?M, a, b?G. In this paper, we prove that a mapping T: M ? M is a centralizer if and only if it is a centralizing left centralizer. We also show that if T and S are left centralizers of M such that T(x)a x + x a S(x)?Z(M) (the center of M) for all x?M, a?G, then both T and S are centralizers.
Keywords: Semiprime G-ring; Left (right) centralizer; Centralizer; Commuting mapping; Centralizing mapping: Extended centroid.
© 2012 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved.
doi: http://dx.doi.org/10.3329/jsr.v4i2.8691 J. Sci. Res. 4 (2), 349-356 (2012)
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