On centrally-extended multiplicative (generalized)-$(\alpha,\beta)$-derivations in semiprime rings

Abstract

Let R be a ring with center Z and α, β and d mappings of R. A mapping F of R is called a centrally-extended multiplicative (generalized)-(α, β)-derivation associated with d if F (xy) − F (x)α(y) − β(x)d(y) ∈ Z for all x, y ∈ R. The objective of the present paper is to study the following conditions: (i) F (xy) ± β(x)G(y) ∈ Z, (ii) F (xy) ± g(x)α(y) ∈ Z and (iii) F (xy) ± g(y)α(x) ∈ Z for all x, y in some appropriate subsets of R, where G is a multiplicative (generalized)-(α, β)-derivation of R associated with the map g on R.

On centrally extended Jordan derivations and related maps in rings

“…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

“…in(21), we see that 6[Π(s), s]s 2 + 6[Π(s), s]s 2 ∈ Z. Thus, 12[Π(s), s]s 2 ∈ Z, and so, [Π(s), s]s 2 ∈ Z. Using(20) in the last relation, we find that [Π(s), s] = 0 or s 2 ∈ Z. Suppose that s 2 ∈ Z; the same as in the proof of Theorem 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].…”

“…By linearizing(20), we see that[Π(s), s 1 ] + [Π(s 1 ), s] = 0 (23)for all s, s 1 ∈ S. Taking s 1 by s • h in(23), where h ∈ H, we find that[Π(s), sh + hs] + [Π(s) • h + s • Π(h), s] = 0. Putting hby s 1 • s in the above relation, where s 1 ∈ S, we conclude that s[Π(s), s 1 s + ss 1 ] + [Π(s), s 1 s + ss 1 ]s + Π(s)[s 1 s + ss 1 , s] + [s 1 s + ss 1 , s]Π(s) + s[Π(s 1 )s + sΠ(s 1 ) + s 1 Π(s) + Π(s)s 1 , s]…”