Asian Journal of Algebra1994-540x2077-2025Asian Network for Scientific Information10.3923/aja.2012.11.20DeyKalyan Kumar PaulAkhil Chandra RakhimovIsamiddin S. 1201251Let 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, α∈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, α,β∈Γ. For the case of semiprime Γ-ring M, we also prove that the mapping σ+τ is a free action if σ and τ are automorphisms of M.]]>Kallman, R.R.,196936781789Murray, F.J. and J. von Neumann,193637116229Neumann, J.V.,19404194161Barnes, E.,196618411422Chakraborty, S. and A.C. Paul,2007N-rings.]]>5728232829Ozturk, M.A. and Y.B. Jun,200015469477Ozturk, M.A. and Y.B. Jun,200125367377