Jordan derivations of unital algebras with idempotents

Abstract

We consider Jordan derivations of a unital algebra A having a nontrivial idempotent. It turns out that on unital algebras there exist Jordan derivations that are not derivations. For this purpose we introduce the term a singular Jordan derivation, which is a proper Jordan derivation of the form that depends on Peirce decomposition of the unital algebra A. Singular Jordan derivations are usually antiderivations. The main result of the paper states that under certain conditions every Jordan derivation of A is the sum of a derivation and a singular Jordan derivation.

Centralizing traces and Lie triple isomorphisms on generalized matrix algebras

“…It was Krylov who initiated the study of linear mappings on generalized matrix algebras from the point of classifying view [47]. Since then many articles are devoted to this topic, and a number of interesting results are obtained (see [12,15,37,38,52,53,54,67,74,75]). Nevertheless, it leaves so much to be desired.…”

Integrations on rings

“…This concept of derivations on rings has also been generalized (as well as restricted) in many ways, for instance, there are inner derivations, Jordan derivations, Lie derivations, (θ, φ)-derivations, generalized derivations, left derivations, and more; for examples see [1,2,4,5,7,8,10,12,13,14,17]. In present paper we also deal with two such functions that are related to a derivation, see Definition 2. for all x, y ∈ R, where x • y = x · y + y · x denotes the Jordan product on R.…”

Jordan Higher Derivations on Trivial Extension Algebras

“…Zhang an Yu [15] proved that every Jordan derivation on a triangular algebra is a derivation. Benkovič andŠirovnik [1] studied Jordan derivations on a unital algebra with a nontrivial idempotent. Li et al [6] investigated the Jordan derivations on a generalized matrix algebra.…”

“…In Section 3, we first show that every trivial generalized matrix algebra is a trivial extension algebra. We then explore the structure of (Jordan) higher derivations of a trivial generalized matrix algebra, intending to arrive at the "higher" version of some results of [6] and [1]. In this respect, we leave a conjecture, to the best of our knowledge, seems to be undecided.…”