Some results on ∗-differential identities in prime rings

Let [Formula: see text] be a prime ring with involution ∗ of the second kind. An additive mapping [Formula: see text] is called generalized derivation if there exists a unique derivation [Formula: see text] such that [Formula: see text] for all [Formula: see text] In this paper, we investigate the structure of [Formula: see text] and describe the possible forms of generalized derivations of [Formula: see text] that satisfy specific ∗-differential identities. Precisely, we study the following situations: (i) [Formula: see text] (ii) [Formula: see text] (iii) [Formula: see text] (iv) [Formula: see text] for all [Formula: see text] Moreover, we construct some examples showing that the restrictions imposed in the hypotheses of our theorems are not redundant.

Generalized g-derivations on prime rings

We introduce the definitions of [Formula: see text]-derivations and generalized [Formula: see text]-derivations on a ring [Formula: see text]. The main objective of the paper is to describe the structure of a prime ring [Formula: see text] in which [Formula: see text]-derivations and generalized [Formula: see text]-derivations satisfy certain algebraic identities with involution ⋆, anti-automorphism and automorphism. Some well-known results concerning derivations, generalized derivations, skew derivations and generalized skew derivations in prime rings, have been generalized to the case of [Formula: see text]-derivations and generalized [Formula: see text]-derivations.

Generalized derivations, quasiderivations and centroids of Bihom-Lie triple system

In this paper, we give some properties of the generalized derivation algebra [Formula: see text] of a Bihom-Lie triple system [Formula: see text]. In particular, we prove that [Formula: see text], the sum of the quasiderivation algebra and the quasicentroid. We also prove that [Formula: see text] can be embedded as derivations in a larger Bihom-Lie triple system.

Some Identities of 3-Prime Near-Rings Involving Jordan Ideals and Left Generalized Derivations

In the current paper, we study the structure of Jordan ideals of a 3-prime near-ring which satisfies some algebraic identities involving left generalized derivations and right centralizers. The limitations imposed in the hypothesis were justified by examples.

Algebras with representable representations

Just like group actions are represented by group automorphisms, Lie algebra actions are represented by derivations: up to isomorphism, a split extension of a Lie algebra $B$ by a Lie algebra $X$ corresponds to a Lie algebra morphism $B\to {\mathit {Der}}(X)$ from $B$ to the Lie algebra ${\mathit {Der}}(X)$ of derivations on $X$ . In this article, we study the question whether the concept of a derivation can be extended to other types of non-associative algebras over a field ${\mathbb {K}}$ , in such a way that these generalized derivations characterize the ${\mathbb {K}}$ -algebra actions. We prove that the answer is no, as soon as the field ${\mathbb {K}}$ is infinite. In fact, we prove a stronger result: already the representability of all abelian actions – which are usually called representations or Beck modules – suffices for this to be true. Thus, we characterize the variety of Lie algebras over an infinite field of characteristic different from $2$ as the only variety of non-associative algebras which is a non-abelian category with representable representations. This emphasizes the unique role played by the Lie algebra of linear endomorphisms $\mathfrak {gl}(V)$ as a representing object for the representations on a vector space $V$ .