On centrally extended Jordan derivations and related maps in rings

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.

Centrally Extended Jordan (∗)-Derivations Centralizing Symmetric or Skew Elements

“…Let A be a ring with involution " * ". Recently, Bhushan et al [17] introduced the notion of CE-Jordan derivation. They established the following result: if A is a non-commutative prime ring, char(A) = 2 with involution " * ", and Π is a CE-Jordan derivation such that [Π(a), a] ∈ Z (resp., [Π(a), a * ] ∈ Z) for all a ∈ A, then Π = 0 or dim C AC ≤ 4.…”

“…for all s ∈ S. Taking s = s in (17), we obtain 4s Π(s ) ∈ Z; so, Π(s ) ∈ Z. Applying the last relation in (17), we see that 2s Π(s) + 2Π(s )s ∈ Z.…”

“…for all s ∈ S. Taking s = s in (17), we obtain 4s Π(s ) ∈ Z; so, Π(s ) ∈ Z. Applying the last relation in (17), we see that 2s Π(s) + 2Π(s )s ∈ Z. This implies that [Π(s), s] = 0 for all s ∈ S. Now, the same as in Subcase (2), we obtain that A satisfies s 4 .…”

“…There has been rising literature investigating centrally extended mappings in rings under various settings; e.g., see [16][17][18][19][20].…”

“…Recently, Bhushan et al [17] introduced centrally extended Jordan derivations, which are a generalization of Jordan derivations and derivations, and they discussed the existence of these mappings in rings. Accordingly, a self-mapping…”