CENTRALIZERS ON PRIME AND SEMIPRIME GAMMA RINGS - Hoque, Md Fazlul; Paul, A.C. (Italian Journal of Pure and Applied Mathematics, 2015)
Let M be a noncommutative 2-torsion free semiprime Γ-ring satisfying a certain assumption and let S and T be left centralizers on M. We prove the following results: (i) If [S(x),T(x)]αβS(x) + S(x)β[S(x),T(x)]α=0 holds for all x ∈ M and α, β ∈ Γ, then [S(x),T(x)]α=0. (ii) If S ≠ 0 (T ≠ 0), then there exists λ ∈ C, (the extended centroid of M ) such that T=λαS (S=λαT ) for all α ∈ Γ. (iii) Suppose that [[S(x),T(x)]α,S(x)]β =0 holds for all x ∈ M and α, β ∈ Γ. Then [S(x),T(x)]α=0 for all x ∈ M and α ∈ Γ. (iv) If M is a prime Γ-ring satisfying a certain assumption and S ≠ 0 (T ≠ 0), then there exists λ ∈ C, the extended centroid, such that T=λαS (S=λαT).