On Lie ideals with generalized derivations

Let R be a prime ring with characteristic different from 2, let U be a nonzero Lie ideal of R, and let f be a generalized derivation associated with d. We prove the following results: (i) If a ∈ R and [a, f(U)] = 0 then a ∈ Z or d(a) = 0 or U ? Z; (ii) If f ²(U) = 0 then U ? Z; (iii) If u ² ∈ U for all u ∈ U and f acts as a homomorphism or antihomomorphism on U then either d = 0 or U ? Z.     

On approximately higher ring derivations

In this paper, we examine the Hyers-Ulam, the Isac and Rassias-type stability and the Bourgin-type superstability of a functional inequality corresponding to the following functional equation, respectively:

     

On the product of two generalized derivations

Two elements and in a ring determine a generalized derivation on by setting for any in . We characterize when the product is a generalized derivation in the cases when the ring is the algebra of all bounded operators on a Banach space , and when is a -algebra . We use these characterizations to compute the commutant of the range of .

     

Jordan generalized derivations on triangular algebras

Let 𝒜 be a unital algebra and let ? be a unitary 𝒜-bimodule. We consider Jordan generalized derivations mapping from 𝒜 into ?. Our results on unitary algebras are applied to triangular algebras. In particular, we prove that any Jordan generalized derivation of a triangular algebra is a generalized derivation.     

A characterization of generalized derivations of JSL algebras

Let AlgL be a J-subspace lattice algebra on a Banach space X and M be an operator in AlgL. We prove that if δ : AlgL → B(X) is a linear mapping satisfying δ(AB) = δ(A)B + Aδ(B)for all A, B ∈ AlgL with AMB = 0, then δ is a generalized derivation. This result can be applied to atomic Boolean subspace lattice algebras and pentagon subspace lattice algebras.     

On the Hyers-Ulam stability of functional equations connected with additive and quadratic mappings

We investigate some inequalities connected with the Hyers-Ulam stability of three functional equations, which have a solution of the form φ=a+q, where a is an additive mapping and q is a quadratic one.     

On a direct method for proving the Hyers-Ulam stability of functional equations

We study the stability of an equation in a single variable of the form

f(x)=af(h(x))+bf(−h(x))     

Bicommutants and ranges of derivations

Let V be a vector space, V* its dual space and L(V) the algebra of all linear operators on V. For an operator a?∈?L(V) let a* be its adjoint acting on V*, and for a subset R of L(V) let R″ be its bicommutant. If R is a left noetherian subalgebra of L(V), then {a*: a?∈?R}″?=?{a*: a?∈?R″}. When R is singly generated R″ is described precisely. Further, for any two operators a, b?∈?L(V), b?∈?(a)″ if and only if the derivations d _a and d _b satisfy d _b (F(V))???d _a (F(V)), where F(V) is the set of all finite rank operators on V. In this case the inclusion d _b (L(V))???d _a (L(V)) also holds.     

On generalized left derivations in rings and Banach algebras

The purpose of this paper is to establish some results concerning generalized left derivations in rings and Banach algebras. In fact, we prove the following results: Let R be a 2-torsion free semiprime ring, and let \({G: R \longrightarrow R}\) be a generalized Jordan left derivation with associated Jordan left derivation \({\delta: R \longrightarrow R}\). Then every generalized Jordan left derivation is a generalized left derivation on R. This result gives an affirmative answer to the question posed as a remark in Ashraf and Ali (Bull. Korean Math. Soc. 45:253–261, 2008). Also, the study of generalized left derivation has been made which acts as a homomorphism or as an anti-homomorphism on some appropriate subset of the ring R. Further, we introduce the notion of generalized left bi-derivation and prove that if a prime ring R admits a generalized left bi-derivation G with associated left bi-derivation B then either R is commutative or G is a right bi-centralizer (or bi-multiplier) on R. Finally, it is shown that every generalized Jordan left derivation on a semisimple Banach algebra is continuous.     

Some results on stability of extended derivations

In this paper we prove the generalized Hyers-Ulam-Rassias stability of extended derivations on unital Banach algebras associated to a generalized Jensen equation.     

Vanishing and cocentralizing generalized derivations on Lie ideals

Let R be a prime ring of characteristic different from 2, U its right Utumi quotient ring, C its extended centroid and L a not central Lie ideal of R. Suppose that F, G and H are generalized derivations of R, with F≠0, such that F(G(x)x?xH(x)) = 0, for any x∈L. In this paper we describe all possible forms of F, G and H.     

A note on generalized derivations in Banach algebras

In this note we characterize a complex Banach algebra A admitting a generalized derivation g such that the cardinality of the spectrum σ(g(x)) is exactly one for all x∈A.     

Strong commutativity preserving generalized derivations on Lie ideals

We apply elementary matrix computations and the theory of differential identities to prove the following: let R be a prime ring with extended centroid C and L a noncommutative Lie ideal of R. Suppose that f?:?L?→?R is a map and g is a generalized derivation of R such that [f(x),?g(y)]?=?[x,?y] for all x,?y?∈?L. Then there exist a nonzero α?∈?C and a map μ?:?L?→?C such that g(x)?=?αx for all x?∈?R and f(x)?=?α^?1 x?+?μ(x) for all x?∈?L, except when R???M ₂(F), the 2?×?2 matrix ring over a field F.     

Strong commutativity-preserving generalized derivations on semiprime rings

Let R be a ring with a subset S. A mapping of R into itself is called strong commutativitypreserving （scp） on S, if [f（x）, f（y）] = [x, y] for all x, y ∈ S. The main purpose of this paper is to describe the structure of the generalized derivations which are scp on some ideals and right ideals of a prime ring, respectively. The semiprime case is also considered.