Nonlinear Lie triple higher derivation on triangular algebras

Abstract Let be the triangular algebra consisting of unital algebras A and B over a commutative ring R with identity 1 and M be a unital (A; B)-bimodule. An additive subgroup L of A is said to be a Lie ideal of A if [L;A] ⊆ L. A non-central square closed Lie ideal L of A is known as an admissible Lie ideal. The main result of the present paper states that under certain restrictions on A, every generalized Jordan triple higher derivation of L into A is a generalized higher derivation of L into A.

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Generalized Jordan triple (σ,τ)-higher derivation on triangular algebras

Filomat ◽

10.2298/fil1908285 ◽

2019 ◽

Vol 33 (8) ◽

pp. 2285-2294

Author(s):

Mohammad Ashraf ◽

Aisha Jabeen ◽

Mohd Akhtar

Keyword(s):

Commutative Ring ◽

Triangular Algebra ◽

Higher Derivation ◽

Triangular Algebras

Let R be a commutative ring with unity, U = Tri(A,M,B) be a triangular algebra consisting of unital algebras A,B and (A,B)-bimodule M which is faithful as a left A-module and also as a right B-module. Let ? and ? be two automorphisms of U. A family ? = {?n}n?N of R-linear mappings ?n : U ? U is said to be a generalized Jordan triple (?,?)-higher derivation on A if there exists a Jordan triple (?,?)-higher derivation D = {dn}n?N on U such that ?0 = IU, the identity map of U and ?n(XYX) = ?i+j+k=n ?i(?n-i(X))dj(?k?i(Y))dk(?n-k(X)) holds for all X,Y ? U and each n ? N. In this article, we study generalized Jordan triple (?,?)-higher derivation on A and prove that every generalized Jordan triple (?,?)-higher derivation on U is a generalized (?,?)-higher derivation on U.

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Lie higher derivations on triangular algebras revisited

Filomat ◽

10.2298/fil1612187m ◽

2016 ◽

Vol 30 (12) ◽

pp. 3187-3194 ◽

Cited By ~ 1

Author(s):

F. Moafian ◽

Ebrahimi Vishki

Keyword(s):

Direct Proof ◽

Acta Math ◽

Multilinear Algebra ◽

Triangular Algebra ◽

Higher Derivation ◽

Linear Multilinear Algebra ◽

Triangular Algebras

Motivated by the extensive works of W.-S. Cheung [Linear Multilinear Algebra, 51 (2003), 299-310] and X.F. Qi [Acta Math. Sinica, English Series, 29 (2013), 1007-1018], we present the structure of Lie higher derivations on a triangular algebra explicitly. We then study those conditions under which a Lie higher derivation on a triangular algebra is proper. Our approach provides a direct proof for some known results concerning to the properness of Lie higher derivations on triangular algebras.

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