CENTRALIZERS ON PRIME AND SEMIPRIME GAMMA RINGS

Abstract

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).