Linear maps on block upper triangular matrix algebras behaving like Jordan derivations through commutative zero products

Abstract

Let T = T (n 1 ,n 2 ,••• ,n k) ⊆ M n (C) be a block upper triangular matrix algebra and let M be a 2-torsion free unital T-bimodule, where C is a commutative ring. Let Δ : T → M be a C-linear map. We show that if Δ(X)Y + XΔ(Y)+Δ(Y)X +Y Δ(X) = 0 whenever X,Y ∈ T are such that XY = Y X = 0 , then Δ(X) = D(X) + α(X) + XΔ(I) , where D : T → M is a derivation, α : T → M is an antiderivation, I is the identity matrix and Δ(I)X = XΔ(I) for all X ∈ T. We also prove that under some sufficient conditions on T , we have α = 0. As a corollary, we show that under given sufficient conditions, each Jordan derivation Δ : T → M is a derivation and this is an answer to the question raised in [9]. Some previous results are also generalized by our conclusions.

Characterizing Jordan Derivable Maps on Triangular Rings by Local Actions

“…First Brešar in [14] proved that if δ is an additive map on a unital ring R, that contains a nontrivial idempotent and aδ(b) + δ(a)b � 0 for all a, b ∈ R with ab � 0, then δ(a) � τ(a) + ca, where τ is an additive derivation on R and c belongs to the center of R. Following this line of investigation, derivations and Jordan derivations (additive or nonadditive) at zero products or another special pairs of several rings or algebras has been studied and considerable results has been achieved. For instance, see [12,[15][16][17][18][19][20][21] and the references therein.…”