Lie ideals and derivations of exceptional prime rings

A prime ring [Formula: see text] with extended centroid [Formula: see text] is said to be exceptional if both [Formula: see text] and [Formula: see text]. Herstein characterized additive subgroups [Formula: see text] of a nonexceptional simple ring [Formula: see text] satisfying [Formula: see text]. In 1972, Lanski and Montgomery extended Herstein’s theorem to nonexceptional prime rings. In the paper, we first extend Herstein’s theorem to arbitrary simple rings. For the prime case, let [Formula: see text] be an exceptional prime ring with center [Formula: see text]. It is proved that if [Formula: see text] is a noncentral additive subgroup of [Formula: see text] satisfying [Formula: see text] for some nonabelian Lie ideal [Formula: see text] of [Formula: see text], then [Formula: see text] for some nonzero [Formula: see text], and either [Formula: see text] for some [Formula: see text] with [Formula: see text] or [Formula: see text]. Second, we study certain generalized linear identities satisfied by Lie ideals and then completely characterize derivations [Formula: see text] of [Formula: see text] satisfying [Formula: see text] for [Formula: see text] a Lie ideal of [Formula: see text].