Characterization of Lie-type higher derivations of triangular rings

Abstract

Abstract Let 𝔄 {\mathfrak{A}} be a triangular ring and let p n ⁢ ( U 1 , U 2 , … , U n ) {p_{n}(U_{1},U_{2},\dots,U_{n})} denote the ( n - 1 ) {(n-1)} th commutator of elements U 1 , U 2 , … , U n ∈ 𝔄 {U_{1},U_{2},\dots,U_{n}\in\mathfrak{A}} . Suppose that ℕ {\mathbb{N}} is the set of nonnegative integers and 𝔏 = { ξ r } r ∈ ℕ {\mathfrak{L}=\{{\xi_{r}}\}_{r\in\mathbb{N}}} is a sequence of additive mappings on 𝔄 {\mathfrak{A}} such that ξ 0 = i ⁢ d 𝔄 {\xi_{0}=id_{\mathfrak{A}}} , the identity mapping on 𝔄 {\mathfrak{A}} , and for each r ∈ ℕ {r\in\mathbb{N}} , ξ r ⁢ ( p n ⁢ ( U 1 , U 2 , … , U n ) ) = ∑ i 1 + i 2 + ⋯ + i n = r p n ⁢ ( ξ i 1 ⁢ ( U 1 ) , ξ i 2 ⁢ ( U 2 ) , … , ξ i n ⁢ ( U n ) ) {\xi_{r}(p_{n}(U_{1},U_{2},\dots,U_{n}))=\sum_{i_{1}+i_{2}+\cdots+i_{n}=r}p_{n% }(\xi_{i_{1}}(U_{1}),\xi_{i_{2}}(U_{2}),\dots,\xi_{i_{n}}(U_{n}))} for all U 1 , U 2 , … , U n ∈ 𝔄 {U_{1},U_{2},\dots,U_{n}\in\mathfrak{A}} with U 1 ⁢ U 2 ⁢ ⋯ ⁢ U n = 0 {U_{1}U_{2}\cdots U_{n}=0} . In this paper, it is shown that under certain conditions 𝔏 = { ξ r } r ∈ ℕ {\mathfrak{L}=\{{\xi_{r}}\}_{r\in\mathbb{N}}} has the standard form, that is, there exist a higher derivation { d r } r ∈ ℕ {\{{d_{r}}\}_{r\in\mathbb{N}}} on 𝔄 {\mathfrak{A}} and a family { h r } r ∈ ℕ {\{{\mathrm{h}_{r}}\}_{r\in\mathbb{N}}} of additive mappings h r : 𝔄 → 𝒵 ⁢ ( 𝔄 ) {h_{r}:\mathfrak{A}\rightarrow\mathcal{Z}(\mathfrak{A})} satisfying h r ⁢ ( p n ⁢ ( U 1 , U 2 , … , U n ) ) = 0 {h_{r}(p_{n}(U_{1},U_{2},\dots,U_{n}))=0} for all U 1 , U 2 , … , U n ∈ 𝔄 {U_{1},U_{2},\dots,U_{n}\in\mathfrak{A}} with U 1 ⁢ U 2 ⁢ ⋯ ⁢ U n = 0 {U_{1}U_{2}\cdots U_{n}=0} such that for each r ∈ ℕ {r\in\mathbb{N}} , ξ r ⁢ ( U ) = d r ⁢ ( U ) + h r ⁢ ( U ) {\xi_{r}(U)=d_{r}(U)+h_{r}(U)} for all U ∈ 𝔄 {U\in\mathfrak{A}} .