On rings with some kinds of centrally-extended maps

Multiplicative Generalized Reverse ∗ CE-Derivations Acting on Rings with Involution

“…Tey discussed the existence of such map which is not a derivation and gave some commutativity results. In [2], the authors generalized this notion to other kinds of maps and extended some results due to Bell and Daif. Recently, in [3], the authors gave the notion of Jordan CE-derivations and, under some conditions, they proved that every Jordan CE-derivation of a prime ring S is a CE-derivation.…”

On centrally-extended multiplicative (generalized)-$(\alpha,\beta)$-derivations in semiprime rings

“…Let S be a nonempty subset of R and α a mapping of R. If α(xy) = α(x)α(y) or α(xy) = α(y)α(x) for all x, y ∈ S, then we say that α acts as homomorphism or anti-homomorphism on S, respectively. A map f : S → R is said to be α-commuting on S in case [α(x), f (x)] = 0 satisfies for all x ∈ S. We will make some extensive use of the basic commutator identities [5] generalized this notion to the concepts CE-(α, β)-derivation and CE-generalized (α, β)-derivation.…”

“…Recently, Bell and Daif [2] introduced the notion of centrally-extended derivations (CEderivation) on rings. A CE-derivation D of R is a mapping of R such that D(x+y)−D(x)− D(y) ∈ Z and D(xy)−D(x)y−xD(y) ∈ Z, for all x, y in R. Tammam et al [5] generalized this notion to the concepts CE-(α, β)-derivation and CE-generalized (α, β)-derivation.…”