Generalized Jordan Derivations on Semiprime Rings

Abstract

<jats:p>The purpose of this note is to prove the following. Suppose <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1446788719000259_inline1.png" /><jats:tex-math>$\mathfrak{R}$</jats:tex-math></jats:alternatives></jats:inline-formula> is a semiprime unity ring having an idempotent element <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1446788719000259_inline2.png" /><jats:tex-math>$e$</jats:tex-math></jats:alternatives></jats:inline-formula> (<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1446788719000259_inline3.png" /><jats:tex-math>$e\neq 0,~e\neq 1$</jats:tex-math></jats:alternatives></jats:inline-formula>) which satisfies mild conditions. It is shown that every additive generalized Jordan derivation on <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S1446788719000259_inline4.png" /><jats:tex-math>$\mathfrak{R}$</jats:tex-math></jats:alternatives></jats:inline-formula> is a generalized derivation.</jats:p>

Centrally-extended generalized Jordan derivations in rings

“…By Lemma 2.4, R is an order in a central simple algebra of dimension at most 4 over its center or δ = 0. In case δ = 0, from (10) we obtain σ(x)x ∈ C. Using the primeness of R and Brauer's trick, we conclude that either σ = 0 or R is commutative. Clearly, R can not be commutative, therefore we have from (8), F (x) = λx for all x ∈ R. This completes the proof.…”

“…In view of our assumption Z(R) = {0}, therefore we are left with δ(Z(R)) ⊆ Z(R). Polarizing (10), we have…”

“…Moreover, the question "when Jordan derivation is a derivation?" raised by Herstein [12] caused significant work existed in the literature of Jordan mappings in rings (see [6], [10], [12], [13]). In 1991, Bresar [7] introduced the notion of generalized derivation.…”

On ultra algebras

“…In this paper we have introduced Jordan ultra algebras, and then we have the following question: What is the connection between the structure of ultra algebras and Jordan ultra algebras? In addition, the study about map structures on nonassociative algebras has become an area of great interest of pure math in the last years, we can quote some recent works [1,2,3,4,5,6,7]. Therefore other line of research that appears here is to know when a map is additive on nonassociative ultra algebras.…”

ELEMEN IDEMPOTEN DI GELANGGANG Z_n DAN KAITANNYA DENGAN HOMOMORFISMA GELANGGANG

“…Ada beberapa kajian tingkat lanjut mengenai elemen idempoten dari suatu gelanggang [13], [14], [15], tetapi pada makalah ini yang akan dikaji adalah kaitan antara elemen idempoten dengan homomorfisma gelanggang, secara khusus untuk homomorfisma gelanggang dari ke ℤ 𝑛 , dengan m dan n dua bilangan asli yang boleh berbeda.…”