On Jordan Triple Higher Derivable Mappings on Rings

Abstract

Let R be a ring and N be the set of all non-negative integers. A family of maps D = {dn} n∈N is said to be Jordan triple higher derivable if dn(aba) = p+q+r=n dp(a)dq(b)dr(a) holds for all a, b ∈ R, where d0 = IR, (the identity map on R). In this paper, we determine Jordan triple higher derivable map on a ring R, which contains a nontrivial idempotent which is automatically additive. An immediate application of our main result shows that every Jordan triple higher derivable map becomes higher derivation on R.Mathematics Subject Classification. 16W25, 16Y30.

Multiplicative Jordan triple higher semi-derivations on rings and standard operator algebras

“…Daif [5] introduced the definition of a multiplicative derivation of a ring and studied the additivity of such map. In particular, Jing and Lu [7] studied the additivity of multiplicative Jordan triple derivations on rings and Ashraf and Parveen [2] studied the additivity of multiplicative Jordan triple higher derivations on rings. Considering that Ferreira and Marietto [6] introduced the notion of a multiplicative Jordan triple semi-derivation and studied its additivity and inspired by the facts mentioned above and the notions of [1], in this paper we introduce the notions of a multiplicative higher semi-derivation of a ring and multiplicative Jordan triple higher semi-derivation of a ring and present a study on the additivity of the last class.…”

Nonlinear Jordan Derivable Mappings of Generalized Matrix Algebras by Lie Product Square-Zero Elements

“…Lu in [10] showed that every nonlinear Jordan derivable mapping on a 2-torsion free semiprime ring is an additive derivation. Ashraf and Jabeen in [11] showed that every nonlinear Jordan triple derivable mapping on a 2-torsion free triangular algebra is an additive derivation. In particular, Benkovič in [1] proved that every additive Jordan derivation from an upper triangular matrix algebra to its bimodule is a sum of an additive derivation and an additive antiderivation.…”

∗-Lie higher derivable mappings on ∗-rings

“…Also, Li, Chen and Wang [14] obtained the same result for * -Lie derivable mappings on a von Neumann algebras and proved that every * -Lie derivable mapping on a von Neumann algebra with no central abelian projections can be expressed as the sum of an additive * -derivation and a mapping with image in the centre vanishing at commutators. In addition, the characterization of Lie derivations and * -Lie derivations on various algebras are considered in [2], [4], [5], [8], [7], [9], [13], [15], [20], [23]. Very recently, Alkenani et al [1] gave the characterization of * -Lie derivable mappings on * -rings, more precisely they proved the following result: Theorem 1.1.…”