Jordan Higher Triple Left Resp. Right Centralizers of Prime Γ-Rings

hrough this paper we define the higher triple left resp. right centralizers of a Γ-ring Ɠ, and study some properties of Jordan higher triple left resp. right centralizers of Ɠ, addition to we prove that: every Jordan higher triple left resp- right centralizer of a Γ-ring Ɠ is higher triple left resp. right centralizer f Ɠ when Ɠ is a 2-torsion free prime gamma ring. Prime Γ-ring, Higher left centralizer, Higher triple left centralizer, Jordan higher triple left centralizer.

Commutativity of Prime Gamma Rings with Left Centralizers

Let M be a G-ring. If M satisfies the condition (*) xaybz = xbyaz for all x, y, zÎM, a, bÎG, then we investigate commutativity of prime G-rings satisfying certain identities involving left centralizer. Keywords: Prime G-ring; Derivation; Generalized derivation; Left centralizer. © 2014 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved. doi: http://dx.doi.org/10.3329/jsr.v6i1.14872 J. Sci. Res. 6 (1), 69-77 (2014)

On Left Centralizers of Semiprime Γ-Rings

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)

ON GENERALIZED (σ, τ)-BIDERIVATIONS IN RINGS

Let σ, τ be automorphisms of a ring R. In the present paper many concepts related to biadditive mappings of rings, viz. σ-left centralizer traces, symmetric generalized (σ, τ)-biderivations, σ-left bimultipliers and symmetric generalized Jordan (σ, τ)-biderivations are studied. Many results related to these concepts are given. It is established that every symmetric generalized (σ, τ)-biderivation of a prime ring of characteristic different from 2, can be reduced to a σ-left bimultiplier under certain algebraic conditions. Further, it is shown that every symmetric generalized Jordan (σ, τ)-biderivation of a prime ring of characteristic different from 2 is a symmetric generalized (σ, τ)-biderivation.