let  be a banach algebra. let  be linear mappings on . first we demonstrate a theorem concerning the continuity of double derivations; especially that all of -double derivations are continuous on semi-simple banach algebras, in certain case. afterwards we define a new vocabulary called “-higher double derivation” and present a relation between this subject and derivations and finally give some ...

Let R be a prime ring with a right generalized (α, β)derivation f and let a ∈ R. Suppose that af(x)n = 0 for all x ∈ R, where n is a fixed positive integer. Then af(x) = 0 for all x ∈R. In particular, if f is either a regular right generalized (α, β)-derivation or a nonzero generalized (α, β)-derivation, then a = 0. In [13] I. N. Herstein proved that if R is a prime ring and d is an inner deriv...

We classify all the pairs of a commutative associative algebra with an identity element and its finite-dimensional commutative locally-finite derivation subalgebra such that the commutative associative algebra is derivation-simple with respect to the derivation subalgebra over an algebraically closed field with characteristic 0. Such pairs are the fundamental ingredients for constructing genera...