Free Actions on Semiprime Gamma Rings

Let M be a semiprime Г-ring. We study on some mappings related to left centralizers, centralizers, derivations, (σ, τ)-derivations and generalized (σ, τ)-derivations which are free actions on semiprime Г-rings. If φ(x) = T(x)αx – xαT(x) for all x in M, α in Г is a mapping from M into M. Then we show that it is a free action. If F : M →M is a generalized (σ,τ)-derivation with associate (σ, τ)-derivation d, and a in F is a dependent element, then we also show that it is a dependent element of (σ+d). Furthermore, we prove that for centralizer f and a derivation d of a semiprime Г-ring M, φ = d◦f is a free action.