Characterizations of Derivations on Prime Rings

In the present paper, we have studied a (α,α)-symmetric derivations D on semiprime rings and prime rings R, we give some results when R admits a (α,α)-symmetric derivations D to satisfy some conditions on R.(i)D([x,y]n+1) =0 for all x, y∈ R. (ii) [D(xn+1),α(y)] = 0 for all x, y ∈R. (iii) [[D(x),α(x)],α(x)]= 0 for all x ∈R. Where α: R →R is an automorphism mapping.