On centrally extended Jordan derivations and related maps in rings

TL;DR

This paper introduces the concepts of centrally extended Jordan derivations and related maps in rings, proving structural results about rings admitting such derivations, specifically characterizing certain prime rings as orders in central simple algebras.

Contribution

It defines new types of derivations in rings and establishes conditions under which rings with these derivations are orders in central simple algebras.

Findings

Rings with centrally extended Jordan derivations are characterized as orders in central simple algebras.

If a prime ring admits a centrally extended Jordan derivation with specific commutation properties, it is an order in a central simple algebra.

Results apply to rings with involution and 2-torsion free conditions.

Abstract

Let $R$ be a ring and $Z(R)$ be the center of $R.$ The aim of this paper is to define the notions of centrally extended Jordan derivations and centrally extended Jordan $\ast$-derivations, and to prove some results involving these mappings. Precisely, we prove that if a $2$-torsion free noncommutative prime ring $R$ admits a centrally extended Jordan derivation (resp. centrally extended Jordan $\ast$-derivation) $\delta:R\to R$ such that \[ [\delta(x),x]\in Z(R)~~(resp.~~[\delta(x),x^{\ast}]\in Z(R))\text{~for~all~}x\in R, \] where $'\ast'$ is an involution on $R,$ then $R$ is an order in a central simple algebra of dimension at most 4 over its center.