Identities With Generalized Derivations

An Engel condition withb-generalized derivations

“…An additive map d : R → R is called a derivation of R if d(xy) = d(x)y + xd(y) for all x, y ∈ R. For a ∈ R, the map x ∈ R → [a, x] = ax − xa ∈ R defines a derivation of R, which is called the inner derivation of R induced by a. An additive map g : R → R is called a generalized derivation of R if there exists a derivation d of R such that g(xy) = g(x)y + xd(y) for all x, y ∈ R (see [2][3][4]). The derivation d is uniquely determined by g and is called the associated derivation of g.…”

On commutators and derivations in rings

“…The usual definition, as introduced in [4] and then used for instance in [9,12,13], is somewhat more general, but to make the paper less technical we shall use this simplified definition in the present paper; besides, in unital rings these two definitions coincide. A generalized derivation G is said to be inner if there are a, b ∈ A such that G = L a − R b .…”

On generalized derivations satisfying certain identities

“…In [14], T. K. Lee and W. K. Shiue proved a version of Kharchenko's theorem for generalized derivations and presented some results concerning certain identities with generalized derivations. More details about generalized derivations can be found in [10,11,13,14].…”