ANNIHILATOR CONDITIONS OF MULTIPLICATIVE REVERSE DERIVATIONS ON PRIME RINGS

Let R be a ring, a mapping F : R → R together with a mapping d : R → R is called a multiplicative (generalized)-reverse derivation if F(xy) = F(y)x + yd(x) for all x, y ∈ R. The aim of this note is to investigate the commutativity of prime rings admitting multiplicative (generalized)-reverse derivations. Precisely, it is proved that for some nonzero element a in R the conditions: a(F(xy) ± xy) = 0, a(F(x)F(y) ± xy) = 0, a(F(xy) ± F(y)F(x)) = 0, a(F(x)F(y) ± yx) = 0, a(F(xy) ± yx) = 0 are sufficient for the commutativity of R. Moreover, we describe the possible forms of generalized reverse derivations of prime rings.