Commutativity with derivations of semiprime rings

Abstract

Abstract Let R be a 2-torsion free semiprime ring with the centre Z(R), U be a non-zero ideal and d : R → R be a derivation mapping.
Suppose that R admits (1) a derivation d satisfying one of the following conditions: (i) [d(x), d(y)] − [x, y] ∈ Z(R) for all x, y ∈ U , (ii) [d2(x), d2(y)] − [x, y] ∈ Z(R) for all x, y ∈ U , (iii) [d(x)[2], d(y)2] − [x, y] ∈ Z(R) for all x, y ∈ U, (iv) [d(x2), d(y2)] − [x, y] ∈ Z(R) for all x, y ∈ U, (v) [d(x), d(y)] − [x2, y2] ∈ Z(R) for all x, y ∈ U.
(2) a non-zero derivation d satisfying one of the following conditions: (i) d([d(x), d(y)]) − [x, y] ∈ Z(R) for all x, y ∈ U, (ii) d([d(x), d(y)]) + [x, y] ∈ Z(R) for all x, y ∈ U.