Dominik Benkovič

Lie triple derivations of unital algebras with idempotents

Let be a unital algebra with a nontrivial idempotent over a unital commutative ring . We show that under suitable assumptions, every Lie triple derivation on is of the form , where is a derivation of , is a singular Jordan derivation of and is a linear mapping from to its centre that vanishes on . As an application, we characterize Lie triple derivations and Lie derivations on triangular algebras and on matrix algebras.

Jordan generalized derivations on triangular algebras

Let 𝒜 be a unital algebra and let ℳ be a unitary 𝒜-bimodule. We consider Jordan generalized derivations mapping from 𝒜 into ℳ. Our results on unitary algebras are applied to triangular algebras. In particular, we prove that any Jordan generalized derivation of a triangular algebra is a generalized derivation.

Lie Triple Derivations on Triangular Matrices

Let be the algebra of all n × n upper triangular matrices over a commutative unital ring , and let be a 2-torsion free unital -bimodule. We show that every Lie triple derivation is a sum of a standard Lie derivation and an antiderivation.

Jordan homomorphisms on triangular matrices

Let be the algebra of all n × n upper triangular matrices over a commutative unital ring . We describe the structure of Jordan homomorphisms from into an arbitrary algebra over . As an application a new proof of our recent result on Jordan derivations on is obtained.

Jordan σ-derivations of triangular algebras

We consider the problem of describing the form Jordan -derivations of a triangular algebra . The main result states that every Jordan -derivation of is of the form , where is a -derivation of and is a special mapping of . We search for sufficient conditions on a triangular algebra, such that . In particular, any Jordan -derivation of a nest algebra is a -derivation and any Jordan -derivation of an upper triangular matrix algebra , where is a commutative unital algebra, is a -derivation.

Generalized skew derivations on triangular algebras determined by action on zero products

ABSTRACT For a triangular algebra 𝒜 and an automorphism σ of 𝒜, we describe linear maps F,G:𝒜→𝒜 satisfying F(x)y+σ(x)G(y) = 0 whenever x,y∈𝒜 are such that xy = 0. In particular, when 𝒜 is a zero product determined triangular algebra, maps F and G satisfying the above condition are generalized skew derivations of the form F(x) = F(1)x+D(x) and G(x) = σ(x)G(1)+D(x) for all x∈𝒜, where D:𝒜→𝒜 is a skew derivation. When 𝒜 is not zero product determined, we show that there are also nonstandard solutions for maps F and G.