ON NILPOTENT DERIVATIONS OF SEMIPRIME RINGS

In this paper we study nilpotent derivations of semiprime rings. An associative derivation d: R → R is an additive mapping on a ring R satisfying d(xy) = d(x) y + xd(y) for all x, y ε{lunate} R. A derivation d: R → R is called inner if d= ad x for some x ε{lunate} R, where ad x(y) = xy - yx. It is proved that for a semiprime ring R, a nilpotent derivation d (with index of nilpotency depending on characteristic) has an extension to the inner derivation and is induced by a nilpotent element of the endomorphism ring End(IR, IR), where I is an essential ideal of R. This is a generalization of some known results due to Kharchenko, Martindale, Chung, and others. © 1992.