Derivations characterized by monomials x2n in prime rings

Abstract

Let [Formula: see text] be a prime ring of [Formula: see text] or [Formula: see text] and let [Formula: see text] be an additive map such that [Formula: see text] for all [Formula: see text], where [Formula: see text] is a positive integer and [Formula: see text] is the maximal symmetric ring of quotients of [Formula: see text]. It is shown that there exist a derivation [Formula: see text] and an additive map [Formula: see text] with [Formula: see text] for all [Formula: see text], such that [Formula: see text]. This result is a natural generalization of the classic theorem of Herstein for Jordan derivations on prime rings. Moreover, it gives a purely algebraic version of the theorem recently obtained by Kosi-Ulbl, Rodriguez and Vukman for standard operator algebras on Banach spaces.