Jordan Left Derivation and Centralizer on Skew Matrix Gamma Ring

We define skew matrix gamma ring and describe the constitution of Jordan left centralizers and derivations on skew matrix gamma ring on a -ring. We also show the properties of these concepts.

Left Derivable Maps at Non-Trivial Idempotents on Nest Algebras

Let Alg 𝒩 be a nest algebra associated with the nest 𝒩 on a (real or complex) Banach space 𝕏. Suppose that there exists a non-trivial idempotent P ∈ Alg 𝒩 with range P (𝕏) ∈ 𝒩, and δ : Alg 𝒩 → Alg 𝒩 is a continuous linear mapping (generalized) left derivable at P, i.e. δ (ab) = aδ (b) + bδ (a) (δ (ab) = aδ(b) + bδ(a) − baδ(I)) for any a, b ∈ Alg 𝒩 with ab = P, where I is the identity element of Alg 𝒩. We show that is a (generalized) Jordan left derivation. Moreover, in a strongly operator topology we characterize continuous linear maps on some nest algebras Alg 𝒩 with the property that δ (P ) = 2Pδ (P ) or δ (P ) = 2P δ (P ) − Pδ (I) for every idempotent P in Alg 𝒩.

The image of Jordan left derivations on algebras

Let A be an algebra, and let I be a semiprime ideal of A. Suppose thatd : A → A is a Jordan left derivation such that d(I) ⊆ I.We prove that if dim{d(a)+I : a ⋲ A} ≤ 1, then d(A) ⊆ I. Additionally, we consider several consequences of this result.

Characterizations of linear mappings through zero products or zero Jordan products

Let $\mathcal{A}$ be a unital algebra and $\mathcal{M}$ be a unital $\mathcal{A}$-bimodule. A characterization of generalized derivations and generalized Jordan derivations from $\mathcal{A}$ into $\mathcal{M}$, through zero products or zero Jordan products, is given. Suppose that $\mathcal{M}$ is a unital left $\mathcal{A}$-module. It is investigated when a linear mapping from $\mathcal{A}$ into $\mathcal{M}$ is a Jordan left derivation under certain conditions. It is also studied whether an algebra with a nontrivial idempotent is zero Jordan product determined, and Jordan homomorphisms, Lie homomorphisms and Lie derivations on zero Jordan product determined algebras are characterized.

On Generalized Derivations ofBCI-Algebras and Their Properties

We introduce the concept of(f,g)-derivations ofBCI-algebras and we investigate some fundamental properties and establish some results on(f,g)-derivations. Also, we treat to generalization of right derivation and left derivation ofBCI-algebras and consider some related properties.

Generalized Jordan left derivations in rings with involution

In the present paper we study generalized left derivations on Lie ideals of rings with involution. Some of our results extend other ones proven previously just for the action of generalized left derivations on the whole ring. Furthermore, we prove that every generalized Jordan left derivation on a 2-torsion free *-prime ring with involution is a generalized left derivation.

Jordan left (?,?) -derivations Of ?-prime rings

It was known that every left (?,?) -derivation is a Jordan left (?,?) – derivation on ?-prime rings but the converse need not be true. In this paper we give conditions to the converse to be true.