Let R be an associative ring. An additive mapping F: R→ R is called a generalized
derivation if there exists a derivation d: R→ R such that F (xy)= F (x) y+ xd (y) holds for all x,
y∈ R. In this paper, we explore the commutativity of the ring R satisfying any one of the
following conditions:(i) d (x)∘ F (y)= 0,(ii)[d (x), F (y)]= 0,(iii) d (x)∘ F (y)= x∘ y,(iv) d (x)∘ F
(y)+ x∘ y= 0,(v) d (x) F (y)-xy∈ Z (R),(vi) d (x) F (y)+ xy∈ Z (R),(vii)[d (x), F (y)]=[x, y] and
(viii)[d (x), F (y)]+[x, y]= 0, for all x, y in some appropriate subset of the ring R.

… In this paper, we extend the notion of a generalized derivation F associated with
derivation d to two generalized derivations F and G associated with the same derivation d,
as a new idea, to obtain the commutativity of prime rings under certain conditions. …
Furthermore, Bresar and Vukman [5] studied the notions of a ∗-derivation and a Jordan ∗-derivation
of R. In this paper, we extend the notion of a generalized derivation F associated with
derivation d to two generalized derivations F and G associated with the same derivation d …