Lie Ideals and Jordan Triple Derivations in Rings

An additive mapping d : R —> R is said to be a Jordan derivation if d{x'^) — d{x)x + xd{x) holds for all x £ R. Trivially, every derivation is a Jordan derivation but there exists example which show that notion of Jordan derivation is some what different form derivation (see Example 1.3.3 ). One can verify that a Jordan derivation in an associative ring i? is a derivation on the Jordan ring under the induced Jordan multiplication. Note that the definition of Jordan derivation presented in the vî ork of Herstein is not as the given above. In fact, Herstein constructed, starting from the ring R, a new ring, namely the Jordan ring of R, defining the new product as aob ~ ab+ba, for any a,b E R. Clearly, this new product is well-defined and it can be early verified that (/?.,+, o) is a ring. So, an additive mapping d, from the Jordan ring into itself, is said by Herstein to be a Jordan derivation, if d{aob) = d{a) ob + aod{b) for every a, 6 e R. However, in the year 1957, Herstein proved a classical result in this direction which becomes a jumping point for many workers latter. The result to which we refer is namely, if i? is a prime ring of characteristic different from 2, then any Jordan derivation is an ordinary derivation of R. An alternative brief proof of this result was later given by Breasar and Vukman [25]. This result was extended to 2-torsion free semiprime rings by Cusack [31] and subsequently, Breasar [21] provided