Abstract: The present paper presents the concepts of group graded Γ-rings and graded Γ-ideals on the basis of the concept of Γ-rings and the obtained some important properties of group graded Γ-rings. By proving Levitizki’s theorem, Xie’s theorem and Herstein-Small’s theorem, e tc., and defining P_G-radical, WP_G-radical and QP_{G-radical, we draw the conclusions which are already true in Γ-rings.}

Key words: G-graded Γ-ring, nil-nilpotent graded Γ -ideal, nil graded Γ-ideal, P_G-radical