Dependent Elements in Prime and Semiprime Gamma Rings

Let M be a Γ-ring. An element a∈M is called a dependent element on a mapping F: M→M if F(x)αa = aαx holds for all x∈M, a∈Γ. In this study, we determine the characterizations of dependent elements on certain mappings on prime and semiprime Γ-rings by taking a certain assumption xαyβz = xβyαz for all x,y,z∈M, αβ∈Γ. For the case of semiprime Γ-ring M, we also prove that the mapping s+t is a free action if s and t are automorphisms of M.