A note on derivations with power central values on a Lie ideal

Generalized derivations with power values on Lie ideals in rings and Banach algebras

“…The first result we mentioned is due to Herstein [15], who prove that if R is a prime ring and d is a nonzero derivation of R such that d(x) n ∈ Z for all x ∈ R for a fixed integer m ≥ 1, then R satisfies s 4 , the standard identity in four variables. The Herstein's result was extended to the case of Lie ideals of prime rings by Bergen and Carini in [4]. They proved that if R is a prime ring of characteristic not 2 and if d is a nonzero derivation of R satisfying d(u) n ∈ Z for all u in some noncentral Lie ideal of R, then also the same conclusion holds.…”

Annihilators of power values of derivations in prime rings

“…We remark that the Main Theorem provides a common generalization of the following results: [13,Theorem], [2,Theorem], [3,Theorem 11, [4, Theorem 31, 117, Theorem 51 and [23,Theorem 11. To proceed the proof of the Main Theorem, we first fix some notation. Let S be a ring.…”

Generalized derivations with central values on lie ideals LIE IDEALS

“…Recall that an additive map d : R → R is called derivation if d(xy) = d(x)y + xd(y), for all x, y ∈ R. Many results in literature indicate that global structure of a prime(semiprime) ring R is often lightly connected to the behaviour of additive mappings defined on R. A well-known result of Herstein [6] stated that if d is a nonzero derivation of a prime ring R such that d(x) n ∈ Z(R) for all x ∈ R, then R satisfies S 4 , the standard identity in four variables. Herstein's result was extended to the case of Lie ideals of prime rings by Bergen and Carini [2]. Some articles was studied derivation with central values on Lie ideals [4,10].…”

“…(see[8, Lemma 2] and[3, Lemma 1]). Let R be a prime ring of charR = 2, L be a noncentral Lie ideal of R and I be the ideal of R generated by [L, L].…”