Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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Characterizations of Lie Triple Higher Derivations of Triangular Algebras by Local Actions

  • Received : 2019.04.24
  • Accepted : 2020.02.25
  • Published : 2020.12.31
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Abstract

Let ℕ be the set of nonnegative integers and 𝕬 be a 2-torsion free triangular algebra over a commutative ring ℛ. In the present paper, under some lenient assumptions on 𝕬, it is proved that if Δ = {𝛿n}n∈ℕ is a sequence of ℛ-linear mappings 𝛿n : 𝕬 → 𝕬 satisfying ${\delta}_n([[x,\;y],\;z])\;=\;\displaystyle\sum_{i+j+k=n}\;[[{\delta}_i(x),\;{\delta}_j(y)],\;{\delta}_k(z)]$ for all x, y, z ∈ 𝕬 with xy = 0 (resp. xy = p, where p is a nontrivial idempotent of 𝕬), then for each n ∈ ℕ, 𝛿n = dn + 𝜏n; where dn : 𝕬 → 𝕬 is ℛ-linear mapping satisfying $d_n(xy)\;=\;\displaystyle\sum_{i+j=n}\;d_i(x)d_j(y)$ for all x, y ∈ 𝕬, i.e. 𝒟 = {dn}n∈ℕ is a higher derivation on 𝕬 and 𝜏n : 𝕬 → Z(𝕬) (where Z(𝕬) is the center of 𝕬) is an ℛ-linear map vanishing at every second commutator [[x, y], z] with xy = 0 (resp. xy = p).

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Acknowledgement

The authors are indebted to the referee for his/her helpful comments and suggestions.

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