Centrally-extended generalized *-derivations on rings with involution

On centrally extended Jordan derivations and related maps in rings

“…Let R be a ring with involution ′ * ′ . In [14], El-Deken and Nabiel introduced the notion of centrally extended * -derivation and investigated the case when centrally extended *derivations are * -derivations. More specifically, they established the following result: If R is a semiprime * -ring with no nonzero central ideals, then every centrally extended * -derivation d on R is a * -derivation.…”

“…There has been a rising literature on the investigation of centrally extended mappings in rings under various settings; for e.g. see [6], [14], [15], [23]. Continuing in this line of investigation, in this paper we introduce centrally extended Jordan derivations and give examples to show the existence of these maps in a 2-torsion free prime rings.…”

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

“…and hence, Π(s 2 )s = sΠ(s 2 ); then, by using the previous expression in(19), we have Π(s 2 )sx − Π(s 2 )xs + sxΠ(s 2 ) − xΠ(s 2 )s = 0. Replacing x by xy in the last equation, right multiplying it by y, and then subtracting them, where y ∈ A, we see that −Π(s 2 )x[y, s] + sx[y, Π(s 2 )] − x[y, Π(s 2 )s] = 0.…”

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