Abstract

Let R be a semiprime ring with extended centroid C and with maximal left ring of quotients Qml(R). An additive map D:R→Qml(R) is called a Jordan triple derivation if D(xyx)=D(x)yx+xD(y)x+xyD(x) for all x,y∈R. In 1957, Herstein proved that a Jordan triple derivation, which is also a Jordan derivation, of a noncommutative prime ring of characteristic 2, must be a derivation. In 1989, Brešar proved that any Jordan triple derivation of a 2-torsion free semiprime ring is a derivation. In the article, we give a complete characterization of Jordan triple derivations of arbitrary semiprime rings. To get such a characterization we first show that, in some sense, an additive map T:R→Qml(R) satisfying T(xyx)=xT(y)x for all x,y∈R can be realized as a centralizer with only an exceptional case that 2R=0 and R is commutative.