Characterizations of generalized Lie $n$-higher derivations on certain triangular algebras

1.   Introduction

Let $R$ be a commutative ring with unity and $A$ be an algebra over $R$. Let $N$ be the set of nonnegative integers and $L = \{L_r\}_{r \in N}$ be a sequence of $R$-linear mappings $L_r: A \rightarrow A$ such that $L_0 = Id_A$. If $L_r(x y) = \sum\limits_{i+j = r} L_i(x) L_j(y)$ holds for all $x, y \in A$, then $L$ is said to be a higher derivation. If $L_r([x, y]) = \sum\limits_{i+j = r}[L_i(x), L_j(y)]$ holds for all $x, y \in A$, then $L$ is said to be a Lie higher derivation. If $L_r(P_n(x_1, x_2, \ldots, x_n)) = \sum\limits_{i_1+i_2+\cdots+i_n = r} P_n(L_{i_1}(x_1), L_{i_2}(x_2), \ldots, L_{i_n}(x_n))$ holds for all $x_1, x_2, \ldots, x_n \in A$, then $L$ is said to be a Lie $n$-higher derivation, where $p_n(x_1, x_2, \ldots, x_n)$ is the ($n-1$)-th commutator.

In 2003, Cheung ^[1] studied Lie derivations on triangular algebras and provided sufficient conditions under which every Lie derivation is the sum of a derivation and a linear mapping into its center. Yu and Zhang ^[2] extended the above result to nonlinear Lie derivations on triangular algebras in 2010. In 2012, Xiao and Wei ^[3] studied nonlinear Lie higher derivations $L = \{L_r\}_{r \in N}$ on triangular algebras and obtained, under certain conditions, that each component $L_r$ can be expressed as a sum of an additive higher derivation and a nonlinear mapping that vanishes on all commutators. Qi ^[4] characterized Lie higher derivations on triangular algebras by acting on zero products and acting on idempotent products. In 2023, Ashraf et al. ^[5] described Lie-type higher derivations of triangular rings by acting on zero products. In 2011, Benkovič ^[6] studied generalized Lie derivations on triangular algebras. Subsequently, Lin ^[7] investigated generalized Lie $n$-derivations on triangular algebras based on the above results. Benkovič ^[8] revisited the study of generalized Lie $n$-derivations by employing commuting and centralizing mappings on triangular algebras. Ashraf and Jabeen ^[9] considered the nonlinear generalized Lie triple higher derivation $L = \{L_r\}_{r \in N}$ on triangular algebras and proved that, under certain assumptions, each component $L_r$ can be expressed through an additive generalized higher derivation and a nonlinear mapping that annihilates on all second commutators.

These observations motivate us to investigate the nonlinear generalized Lie $n$-higher derivations on triangular algebras.

2.   Preliminaries

Let $A$ and $B$ be unital algebras, and let $M$ be a unital $(A, B)$-bi-module, which is faithful as a left $A$-module and also as a right $B$-module. The set

is an associative algebra under the usual matrix operations. Let $1_A$ and $1_B$ be identities of the algebras $A$ and $B$, respectively, then $I = \begin{pmatrix} 1_A & 0 \\ 0 & 1_B \\ \end{pmatrix}$ is the identity of the triangular algebra $\mathcal{U}$. Throughout this paper, we shall use the following notation:

Accordingly, $\mathcal{U}$ can be written as $\mathcal{U} = P\mathcal{U}P\oplus P\mathcal{U}Q \oplus Q\mathcal{U}Q$, where $P\mathcal{U}P$ is a sub-algebra of $\mathcal{U}$ isomorphic to $A$, $Q\mathcal{U}Q$ is a sub-algebra of $\mathcal{U}$ isomorphic to $B$, and $P\mathcal{U}Q$ is a $(P\mathcal{U}P, Q\mathcal{U}Q)$-bi-module isomorphic to the bi-module $M$. In order to simplify, we are going to employ the following symbols $\mathcal{U}_{11} = P\mathcal{U}P$, $\mathcal{U}_{12} = P\mathcal{U}Q$, $\mathcal{U}_{22} = Q\mathcal{U}Q$. Define two natural projections $\pi_{\mathcal{U}_{11}} : \mathcal{U} \rightarrow \mathcal{U}_{11}$ and $\pi_{\mathcal{U}_{22}} : \mathcal{U} \rightarrow \mathcal{U}_{22}$ by $\pi_{\mathcal{U}_{11}}(U_{11}+U_{12}+U_{22}) = U_{11}$ and $\pi_{\mathcal{U}_{22}}(U_{11}+U_{12}+U_{22}) = U_{22}$, where $U_{11}\in\mathcal{U}_{11}, U_{12}\in \mathcal{U}_{12}, U_{22}\in \mathcal{U}_{22}$. It is easy to see that $\pi_{\mathcal{U}_{11}}(Z(\mathcal{U}))$ is a sub-algebra of $Z(\mathcal{U}_{11})$ and that $\pi_{\mathcal{U}_{22}}(Z(\mathcal{U}))$ is a sub-algebra of $Z(\mathcal{U}_{22})$. According to [10, Proposition 3], there exists a unique algebraic isomorphism $\tau: \pi_{\mathcal{U}_{11}}(Z(\mathcal{U})) \rightarrow \pi_{\mathcal{U}_{22}}(Z(\mathcal{U}))$ such that $U_{11}U_{12} = U_{12}\tau(U_{11})$ for all $U_{11}\in \pi_{\mathcal{U}_{11}}(Z(\mathcal{U})), U_{12}\in \mathcal{U}_{12}$. Benkovič ^[11] proposed the following condition for studying Lie $n$-derivations on triangular algebras.

Suppose that $A$ is an associative algebra such that for each $a\in A$,

The above condition (2.1) is very important for studying mappings on triangular algebras, and at the same time he provided the following two remarks.

Remark 2.1. [11, Remark 2.1] Let $\mathcal{U}$ be a triangular ring. For any $U\in \mathcal{U}$ and for any integer $n\geq 2$, we have

Remark 2.2. [11, Remark 2.2] Let $U\in \mathcal{U}$. If $[U, P\mathcal{U}Q] = 0$, then $PUP + QUQ \in Z(\mathcal{U})$.

The concepts outlined below are necessary for this paper.

Let $A$ be an associative algebra and $Z(A)$ be the center of $A$. A mapping $F:A\rightarrow A$ is said to be additive modulo $Z(A)$ if $F(x+y)-F(x)-F(y)\in Z(A)$ for all $x, y \in A$. If $nx = 0$ implies $x = 0$, for some positive integer $n\in N$ and arbitrary $x\in A$, then $A$ is called $n$-torsion free.

We denote the following sequence of polynomials:

The polynomial $p_n(x_1, x_2, \ldots, x_n)$ is said to be an ($n-1$)-th commutator ($n\geq 2$).

Let $N$ be the set of nonnegative integers and $G = \{G_r\}_{r \in N}$ be a sequence of nonlinear mappings $G_r: \mathcal{U} \rightarrow \mathcal{U}$ such that $G_0 = Id_\mathcal{U}$. Then, for all $x_1, x_2, \ldots, x_n \in \mathcal{U}$, $G$ is said to be a:

(ⅰ) Nonlinear generalized higher derivation on $\mathcal{U}$ if there exists a higher derivation $L = \{L_r\}_{r \in N}$ such that $L_0 = Id_\mathcal{U}$ and

(ⅱ) Nonlinear generalized Lie higher derivation on $\mathcal{U}$ if there exists a Lie higher derivation $L = \{L_r\}_{r \in N}$ such that $L_0 = Id_\mathcal{U}$ and

(ⅲ) Nonlinear generalized Lie $n$-higher derivation on $\mathcal{U}$ if there exists a Lie $n$-higher derivation $L = \left\{L_r\right\}_{r \in N}$ such that $L_0 = Id_\mathcal{U}$ and

If $G$ is a nonlinear generalized Lie $n$-higher derivation, then when $r = 1$, $G_1$ is in fact a nonlinear generalized Lie $n$-derivation, i.e., there exists a Lie $n$-derivation $L_1$ such that

for all $x_1, x_2, \ldots, x_n \in \mathcal{U}$. In this case, according to [5, Theorem 3.1], there exist a derivation $d_1: \mathcal{U} \rightarrow \mathcal{U}$ and a mapping $\tau_1: \mathcal{U} \rightarrow Z(\mathcal{U})$ satisfying $\tau_1(P_n(x_1, x_2, \ldots, x_n)) = 0$ such that $L_1(x) = d_1(x)+\tau_1(x)$ for all $x \in \mathcal{U}$. Moreover, $L_1$ and $d_1$ satisfy the following properties:

Although Lin also used the condition $PG(x)Q = 0$ to describe the nonlinear generalized Lie $n$-derivations on triangular algebras, we will characterize the nonlinear generalized Lie $n$-derivations in a new way using the above result. Furthermore, this paper will use the method of induction on $r$ to study the generalized Lie $n$-higher derivations, and the generalized Lie $n$-derivations are precisely the case of $r = 1$.

In Section 2, we present the preliminaries and tools that are necessary. Section 3 investigates the generalized Lie $n$-derivations in a new manner. Subsequently, the generalized Lie $n$-higher derivations were studied through induction on $r$ in Section 4.

3.   Nonlinear generalized Lie $n$-derivations

This section will prove that a nonlinear generalized Lie $n$-derivation $G_1$ of the triangular algebras $\mathcal{U}$ can be expressed as the sum of an additive generalized derivation and a nonlinear mapping vanishing on all ($n-1$)-th commutators on $\mathcal{U}$. To start, we will provide the following lemma.

Lemma 3.1. Let $\mathcal{U} = \mathcal{U}_{11}+\mathcal{U}_{12}+\mathcal{U}_{22}$ be a triangular algebra. Suppose that $\mathcal{U}$ satisfies the following conditions:

(i) $\pi_{\mathcal{U}_{11}}(Z(\mathcal{U})) = Z(\mathcal{U}_{11})$ and $\pi_{\mathcal{U}_{22}}(Z(\mathcal{U})) = Z(\mathcal{U}_{22})$;

(ii) $\mathcal{U}_{11}$ or $\mathcal{U}_{22}$ satisfies (2.1).

If $G_1$ is a nonlinear generalized Lie $n$-derivation on $\mathcal{U}$, then $G_1 = \omega_1+f_1$, where $\omega_1:\mathcal{U} \rightarrow \mathcal{U}$ is additive modulo $Z(\mathcal{U})$ and $f_1:\mathcal{U} \rightarrow Z(\mathcal{U})$ satisfies $f_1(\mathcal{U}_{12}) = \{0\}$.

Proof. The definition of $G_1$ implies that

Therefore,

Since $P_n(U_{11}, U_{22}, U_{12}, Q, \ldots, Q) = 0$, it follows that

It follows from $L_1(\mathcal{U}_{22}) \subseteq \mathcal{U}_{22}+\mathcal{U}_{12}+Z(\mathcal{U})$ that $\left[\left[G_1(U_{11}), U_{22}\right], U_{12}\right] = 0$ for all $U_{11} \in \mathcal{U}_{11}, U_{22} \in \mathcal{U}_{22}, U_{12} \in \mathcal{U}_{12}$. In view of Remark 2.2, we have $Q G_1(U_{11}) Q \in Z(\mathcal{U}_{22})$. Likewise, $P G_1(U_{22}) P \in Z(\mathcal{U}_{11})$ can be derived from $0 = G_1\left(P_n(U_{22}, U_{11}, U_{12}, Q, \ldots, Q)\right)$ for all $U_{22} \in \mathcal{U}_{22}$. In conclusion,

It follows that

Set

for all $U \in \mathcal{U}$. Obviously, $f_1(\mathcal{U})\subseteq Z(\mathcal{U})$, $f_1(\mathcal{U}_{12}) = \{0\}$ and $f_1\left(P_n(U_1, U_2, \ldots, U_n)\right) = 0$ for all $U_1, U_2, \ldots, U_n \in \mathcal{U}$. Define a mapping $\omega_1(U) = G_1(U)-f_1(U)$ for all $U \in \mathcal{U}$. Next, we will prove that $\omega_1:\mathcal{U} \rightarrow \mathcal{U}$ is additive modulo $Z(\mathcal{U})$.

By $U_{12} = P_n(U_{12}, Q, \ldots, Q)$, we obtain

for all $U_{12} \in \mathcal{U}_{12}$. Therefore,

According to (3.1)–(3.3), we arrive at

For any $U_{11}\in \mathcal{U}_{11}$, $U_{12}\in \mathcal{U}_{12}$, on the one hand, it follows from $d_1(Q)\in \mathcal{U}_{12}$ that

and on the other hand, by (3.1), we get

Comparing the above two relations, we obtain

Therefore,

for all $U_{11}\in \mathcal{U}_{11}, U_{12}\in \mathcal{U}_{12}$. Based on

and

and comparing the above two equations, we arrive at

for all $U_{11}\in \mathcal{U}_{11}$, $U_{12}, V_{12}\in \mathcal{U}_{12}.$ It follows from Remark 2.2 that

Similarly, we have

for all $U_{22}\in \mathcal{U}_{22}, U_{12}\in \mathcal{U}_{12}$.

In view of (3.5) and $d_1(V_{12})\in \mathcal{U}_{12}$, we obtain

for all $U_{12}, V_{12} \in \mathcal{U}_{12}$. Since

and

it follows from (3.6) that

for all $U_{12}, V_{12} \in \mathcal{U}_{12}$. On the one hand,

and on the other hand, it follows from (3.7) that

Comparing the above two relations, we have

Since $\mathcal{U}_{12}$ is faithful as a left $\mathcal{U}_{11}$-module, we get

for all $U_{11}, V_{11} \in \mathcal{U}_{11}$. Since $P_n(U_{11}, Q, \ldots, Q) = 0$, we arrive at

That is,

for all $U_{11} \in \mathcal{U}_{11}$. Substituting $U_{11}+V_{11}$ for $U_{11}$ in (3.9), we obtain

Using (3.9) and (3.10), we arrive at

According to (3.8) and (3.11), we find

for all $U_{11}, V_{11} \in \mathcal{U}_{11}$. Using an analogous manner, we show

for all $U_{22}, V_{22} \in \mathcal{U}_{22}$.

For any $U_{11}\in \mathcal{U}_{11}$, $U_{12}\in \mathcal{U}_{12}$, $U_{22}\in \mathcal{U}_{22}$, on the one hand, we have

On the other hand, we get

Comparing the above two relations, we obtain

Therefore,

We have

and

Comparing the above two equations, we arrive at

for all $U_{11}\in \mathcal{U}_{11}$, $U_{12}, V_{12}\in \mathcal{U}_{12}$, $U_{22}\in \mathcal{U}_{22}$. It follows that

for all $U_{11}\in \mathcal{U}_{11}$, $U_{12}\in \mathcal{U}_{12}$, $U_{22}\in \mathcal{U}_{22}$.

Let $U = U_{11}+U_{12}+U_{22}, V = V_{11}+V_{12}+V_{22}$ be two elements in $\mathcal{U}$, where $U_{ij}, V_{ij} \in \mathcal{U}_{ij}$, $1 \leq i \leq j \leq 2$. Then, it follows from (3.7) and (3.12)–(3.14) that

where $\alpha, \beta \in Z(\mathcal{U})$, that is, $\omega_1$ is additive module $Z(\mathcal{U})$. □

Using Lemma 3.1, we present the main theorem of this section.

Theorem 3.2. Let $\mathcal{U} = \mathcal{U}_{11}+\mathcal{U}_{12}+\mathcal{U}_{22}$ be a triangular algebra. Suppose that $\mathcal{U}$ satisfies the following conditions:

(i) $\pi_{\mathcal{U}_{11}}(Z(\mathcal{U})) = Z(\mathcal{U}_{11})$ and $\pi_{\mathcal{U}_{22}}(Z(\mathcal{U})) = Z(\mathcal{U}_{22})$;

(ii) $\mathcal{U}_{11}$ or $\mathcal{U}_{22}$ satisfies (2.1).

If $G_1$ is a nonlinear generalized Lie $n$-derivation on $\mathcal{U}$, then there exists a generalized derivation $\chi_1$ and a nonlinear mapping $h_1$ satisfying $h_1(P_n(U_1, U_2, \ldots, U_n)) = 0$ such that $G_1(U) = \chi_1(U)+h_1(U)$ for all $U, U_1, U_2, \ldots, U_n \in \mathcal{U}$. Additionally, the mappings $G_1$ and $\chi_1$ fulfill the following properties:

Proof. According to Lemma 3.1, we define two mappings $g_1: \mathcal{U} \rightarrow Z(\mathcal{U})$ and $\chi_1: \mathcal{U} \rightarrow \mathcal{U}$ by

and

for all $U = U_{11}+U_{12}+U_{22}\in \mathcal{U}$. It follows that

for all $U_1, U_2, \ldots, U_n \in \mathcal{U}, U_{11}\in \mathcal{U}_{11}, U_{12}\in \mathcal{U}_{12}, U_{22}\in \mathcal{U}_{22}$. Similarly, we have $\chi_1(U_{12}) = \omega_1(U_{12})$ and $\chi_1(U_{22}) = \omega_1(U_{22})$. Therefore,

for all $U_{11}\in \mathcal{U}_{11}, U_{12}\in \mathcal{U}_{12}, U_{22}\in \mathcal{U}_{22}$. In addition, the following properties can be obtained from (3.1)–(3.4):

Let $h_1(U) = g_1(U)+f_1(U)$, then $G_1(U) = \omega_1(U)+f_1(U) = \chi_1(U)+h_1(U)$ for all $U\in \mathcal{U}$. Since $g_1\left(P_n(U_1, U_2, \ldots, U_n)\right) = f_1\left(P_n(U_1, U_2, \ldots, U_n)\right) = 0$, we have $h_1\left(P_n(U_1, U_2, \ldots, U_n)\right) = 0$ for all $U_1, U_2, \ldots, U_n \in \mathcal{U}$. Let $U = U_{11}+U_{12}+U_{22}, V = V_{11}+V_{12}+V_{22}$ be two elements in $\mathcal{U}$, where $U_{ij}, V_{ij} \in \mathcal{U}_{ij}$, $1 \leq i \leq j \leq 2$. Then, it can be inferred from (3.15) and (3.16) that $\chi_1$ is additive. Next, we will prove that $\chi_1$ is a generalized derivation.

According to $U_{11} U_{12} = P_n(U_{11}, U_{12}, Q, \ldots, Q)$, we obtain

for all $U_{11} \in \mathcal{U}_{11}, U_{12} \in \mathcal{U}_{12}$. Similarly, it follows from $U_{12} U_{22} = P_n(U_{12}, U_{22}, Q, \ldots, Q)$ that

for all $U_{12} \in \mathcal{U}_{12}, U_{22} \in \mathcal{U}_{22}$.

In view of (3.17), we get

and

for all $U_{11}, V_{11} \in \mathcal{U}_{11}, U_{12} \in \mathcal{U}_{12}$. Comparing the above equations, we obtain

for all $U_{11}, V_{11} \in \mathcal{U}_{11}, U_{12} \in \mathcal{U}_{12}$. Since $\mathcal{U}_{12}$ is faithful as a left $\mathcal{U}_{11}$-module, we conclude

for all $U_{11}, V_{11} \in \mathcal{U}_{11}$. Replacing $U_{11}$ by $U_{11} V_{11}$ in (3.9), we get

Since $L_1$ is a Lie $n$-derivation, it follows that

This implies that

for all $V_{11} \in \mathcal{U}_{11}$. Multiplying the left side of (3.21) by $U_{11}$ and then combining this with (3.20) yields

for all $U_{11}, V_{11} \in \mathcal{U}_{11}$. Adding (3.19) to (3.22) gives

for all $U_{11}, V_{11} \in \mathcal{U}_{11}$. In a similar manner, we get

for all $U_{22}, V_{22} \in \mathcal{U}_{22}$.

Considering $\chi_1(U_{11}) V_{22}+U_{11} d_1(V_{22}) = G_1 (P_n (U_{11}, V_{22}, Q, \ldots, Q)) = 0$ and (3.17), (3.18), (3.23), (3.24), we can conclude that

for all $U = U_{11}+U_{12}+U_{22}, V = V_{11}+V_{12}+V_{22}\in \mathcal{U}$. □

4.   Nonlinear generalized Lie $n$-higher derivations

This section will study the nonlinear generalized Lie $n$-higher derivations on triangular algebras. First, we present the following result.

Lemma 4.1. [5, Theorem 4.1] Let $\mathcal{U} = \mathcal{U}_{11}+\mathcal{U}_{12}+\mathcal{U}_{22}$ be an ($n-1$)-torsion free triangular ring such that

(i) $\pi_{\mathcal{U}_{11}}(Z(\mathcal{U})) = Z(\mathcal{U}_{11})$ and $\pi_{\mathcal{U}_{22}}(Z(\mathcal{U})) = Z(\mathcal{U}_{22})$;

(ii) $\mathcal{U}_{11}$ or $\mathcal{U}_{22}$ satisfies (2.1).

Let $\{\xi_r\}_{r \in N}$ be a family of additive mappings $\xi_r: \mathcal{U} \rightarrow \mathcal{U}$ such that for each $r \in N$,

for all $U_1, U_2, \ldots, U_n \in \mathcal{U}$ with $U_1 U_2 \cdots U_n = 0$. Then, for each $r \in N, \xi_r = d_r+h_r$, where $\{d_r\}_{r \in N}$ is an additive higher derivation on $\mathcal{U}$ and $\{h_r\}_{r \in N}$ is a family of additive mappings $h_r: \mathcal{U} \rightarrow Z(\mathcal{U})$ vanishing at every ($n-1$)-th commutator $P_n(U_1, U_2, \ldots, U_n)$ with $U_1 U_2 \cdots U_n = 0$. Moreover, $\xi_r$ and $d_r$ satisfy the following properties:

The main theorem of this section is presented below.

Theorem 4.2. Let $\mathcal{U} = \mathcal{U}_{11}+\mathcal{U}_{12}+\mathcal{U}_{22}$ be an ($n-1$)-torsion free triangular algebra satisfying

(i) $\pi_{\mathcal{U}_{11}}(Z(\mathcal{U})) = Z(\mathcal{U}_{11})$ and $\pi_{\mathcal{U}_{22}}(Z(\mathcal{U})) = Z(\mathcal{U}_{22})$;

(ii) $\mathcal{U}_{11}$ or $\mathcal{U}_{22}$ satisfies (2.1).

Let $G = \{G_r\}_{r \in N}$ be a nonlinear generalized Lie $n$-higher derivation on $\mathcal{U}$, then $G_r = \chi_r+h_r$, $r \in N$, where $\{\chi_r\}_{r \in N}$ is an additive generalized higher derivation on $\mathcal{U}$ and $\{h_r\}_{r \in N}$ is a family of nonlinear mappings $h_r: \mathcal{U} \rightarrow Z(\mathcal{U})$ such that $h_r\left(P_n(U_1, U_2, \ldots, U_n)\right) = 0$ for all $U_1, U_2, \ldots, U_n \in \mathcal{U}$.

To prove the theorem, we will employ an inductive approach with respect to the component index $r$. If $r = 1$, then the mapping $G_r: \mathcal{U} \rightarrow \mathcal{U}$ is a nonlinear generalized Lie $n$-derivation. According to Theorem 3.2, this implies the existence of a generalized derivation $\chi_1$ and a nonlinear mapping $h_1$ such that $h_1(P_n(U_1, U_2, \ldots, U_n)) = 0$ and $G_1(U) = \chi_1(U)+h_1(U)$ for all $U, U_1, U_2, \ldots, U_n \in \mathcal{U}$. Furthermore, $G_1$ and $\chi_1$ satisfy the following properties:

We now assume that Theorem 4.2 holds for all $1 < t < r$, $r \in N$. That is, $G_t(U) = \chi_t(U)+h_t(U)$ for all $U \in \mathcal{U}$ and $1 < t < r$, where $\chi_t: \mathcal{U} \rightarrow \mathcal{U}$ is an additive mapping such that $\chi_t(VW) = \sum\limits_{i+j = t} \chi_i(V) d_j(W)$, and $h_t: \mathcal{U} \rightarrow Z(\mathcal{U})$ is a nonlinear mapping such that $h_t(P_n(U_1, U_2, \ldots, U_n)) = 0$ for all $V, W, U_1, U_2, \ldots, U_n \in \mathcal{U}$, where $\{d_j\}_{j \in N}$ is an additive higher derivation on $\mathcal{U}$. In addition, $G_t$ and $\chi_t$ satisfy the following properties:

We will prove that the above properties also hold for $r$. That is, $G_r(U) = \chi_r(U)+h_r(U)$, where $\chi_r$ is an additive mapping satisfying $\chi_r(VW) = \sum\limits_{i+j = r} \chi_i(V) d_j(W)$, and $h_r: \mathcal{U} \rightarrow Z(\mathcal{U})$ is a nonlinear mapping satisfying $h_r(P_n(U_1, U_2, \ldots, U_n)) = 0$ for all $U, V, W, U_1, U_2, \ldots, U_n \in \mathcal{U}.$ To begin, we will introduce the following lemma.

Lemma 4.3. Let $\mathcal{U} = \mathcal{U}_{11}+\mathcal{U}_{12}+\mathcal{U}_{22}$ be a triangular algebra. Suppose that $\mathcal{U}$ satisfies the following conditions:

(i) $\pi_{\mathcal{U}_{11}}(Z(\mathcal{U})) = Z(\mathcal{U}_{11})$ and $\pi_{\mathcal{U}_{22}}(Z(\mathcal{U})) = Z(\mathcal{U}_{22})$;

(ii) $\mathcal{U}_{11}$ or $\mathcal{U}_{22}$ satisfies (2.1),

then the nonlinear mapping $G_r$ mentioned above satisfies $G_r = \omega_r+f_1'$, where $\omega_r:\mathcal{U} \rightarrow \mathcal{U}$ is additive modulo $Z(\mathcal{U})$ and $f_1':\mathcal{U} \rightarrow Z(\mathcal{U})$ satisfies $f_1'(\mathcal{U}_{12}) = \{0\}$.

Proof. Applying the property $C_t$, we get

Again, according to the property $C_t$ and Lemma 4.1, we have

for all $U_{12} \in \mathcal{U}_{12}$. Therefore, $G_r(\mathcal{U}_{12}) \subseteq \mathcal{U}_{12}$.

Using the property $C_t$ and Lemma 4.1, we arrive at

It follows from Remark 2.2 that $Q G_r(U_{11}) Q \in Z(\mathcal{U}_{22})$. Similarly, we can obtain $P G_r(U_{22}) P \in Z(\mathcal{U}_{11})$ from $0 = G_r(P_n(U_{22}, U_{11}, U_{12}, Q, \ldots, Q))$. Hence,

for all $U_{11}\in \mathcal{U}_{11}, U_{22}\in \mathcal{U}_{22}$, that is, $G_r(\mathcal{U}_{i i}) \subseteq \mathcal{U}_{i i}+\mathcal{U}_{12}+Z(\mathcal{U})$ with $i\in\{1, 2\}$. Set

for all $U \in \mathcal{U}$. Clearly, $f_1'(\mathcal{U}_{12}) = \{0\}$, $f_1'(U)\in Z(\mathcal{U})$, and $f_1'\left(P_n(U_1, U_2, \ldots, U_n)\right) = 0$ for all $U, U_1, U_2, \ldots, U_n \in \mathcal{U}$. Define a mapping $\omega_r(U) = G_r(U)-f_1'(U)$ for all $U \in \mathcal{U}$. It is easy to obtain the following relations:

Next, we will prove that $\omega_r:\mathcal{U} \rightarrow \mathcal{U}$ is additive modulo $Z(\mathcal{U})$.

On the one hand, since $d_{i_2}(Q), \ldots, d_{i_n}(Q))\in \mathcal{U}_{12}$, we get

for any $U_{11}\in \mathcal{U}_{11}$, $U_{12}\in \mathcal{U}_{12}$. On the other hand, according to $G_r(0) = 0$, we arrive at

Comparing the above two relations, we obtain

Based on the fact that $\chi_{i_1}$ is additive, we get

Therefore,

In the same way, we can obtain

and

Comparing the above two equations, we have

for all $U_{11}\in \mathcal{U}_{11}$, $U_{12}, V_{12}\in \mathcal{U}_{12}.$ Consequently,

Analogously, we have

Considering

and

we have

Since

and

we can derive from (4.1) that

for all $U_{12}, V_{12} \in \mathcal{U}_{12}$. On the one hand,

and on the other hand, in view of (4.2), we have

As $\chi_{i_1}$ is additive, comparing the two equations above leads us to obtain

Since $\mathcal{U}_{12}$ is faithful as a left $\mathcal{U}_{11}$-module, we can derive from the above expression that

for all $U_{11}, V_{11} \in \mathcal{U}_{11}$. Using $P_n(U_{11}, Q, \ldots, Q) = 0$, we deduce

Replacing $U_{11}$ with $U_{11}+V_{11}$ in (4.4), we arrive at

Again, using (4.4) and (4.5), we obtain

Combining (4.3) and (4.6), we find

for all $U_{11}, V_{11} \in \mathcal{U}_{11}$. In a similar way, we get

for all $U_{22}, V_{22} \in \mathcal{U}_{22}$.

On the one hand, we get

and on the other hand, we have

for all $U_{11}\in \mathcal{U}_{11}$, $U_{12}\in \mathcal{U}_{12}$, $U_{22}\in \mathcal{U}_{22}$. Comparing the above two relations, we arrive at

Hence,

We have

and

Comparing the above two equations, we arrive at

for all $U_{11}\in \mathcal{U}_{11}$, $U_{12}, V_{12}\in \mathcal{U}_{12}$, $U_{22}\in \mathcal{U}_{22}$. It follows that

for all $U_{11}\in \mathcal{U}_{11}$, $U_{12}\in \mathcal{U}_{12}$, $U_{22}\in \mathcal{U}_{22}$. Similar to (3.15), it can be concluded that $\omega_r$ is additive modulo $Z(\mathcal{U})$. □

Proof of Theorem 4.2. Similar to the definitions of $g_1$ and $\chi_1$, the mappings $g_r$ and $\chi_r$ can be defined as

and

and it can be obtained that

for all $U = U_{11}+U_{12}+U_{22}\in \mathcal{U}$. Let $h_r(U) = g_r(U)+f_1'(U)$, then $G_r(U) = \chi_r(U)+h_r(U)$ and $h_r(P_n(U_1, U_2, \ldots, U_n)) = 0$ for all $U, U_1, U_2, \ldots, U_n \in \mathcal{U}$. Additionally, we can conclude from Lemma 4.3 that $\chi_r$ is additive. Next, we will prove that $\chi_r(U V) = \sum_{p+q = r} \chi_p(U) d_q(V)$ for all $U, V \in \mathcal{U}$, where $d_q$ is mentioned in Lemma 4.1.

Given $U_{11} U_{12} = P_n(U_{11}, U_{12}, Q, \ldots, Q)$ and considering the property $C_t$ along with Lemma 4.1, we deduce

for all $U_{11}\in \mathcal{U}_{11}, U_{12}\in \mathcal{U}_{12}$. In a similar manner, we get

for all $U_{12}\in \mathcal{U}_{12}, U_{22}\in \mathcal{U}_{22}$.

According to (4.9) and the property $C_t$, we get

and

Combining the above two relations, we obtain

Since $\mathcal{U}_{12}$ is faithful as a left $\mathcal{U}_{11}$-module, it follows that

for all $U_{11}, V_{11} \in \mathcal{U}_{11}$. Using (4.4), we obtain

Taking into account Lemma 4.1, we conclude

Multiplying the left-hand side of (4.13) by $U_{11}$ and combining it with (4.12), we get

that is,

Hence,

for all $U_{11}, V_{11} \in \mathcal{U}_{11}$. Based on

we deduce

Combining (4.14) and (4.15), and then using

we obtain

Applying (4.11) yields that

for all $U_{11}, V_{11}\in \mathcal{U}_{11}$. Analogously,

for all $U_{22}, V_{22}\in \mathcal{U}_{22}$.

Based on the definition of $G_r$, it is clear that

Therefore, it follows from the property $C_t$ and (4.9), (4.10), (4.16), (4.17) that

for all $U = U_{11}+U_{12}+U_{22}, V = V_{11}+V_{12}+V_{22}\in \mathcal{U}$.

The proof of the theorem is completed. □

5.   Conclusions

We used Ashraf et al.'s results on Lie $n$-derivations to study the nonlinear generalized Lie $n$-derivations, although Lin also used the condition $PG(x)Q = 0$ to describe the nonlinear generalized Lie $n$-derivations on triangular algebras. Based on this, we used an inductive method to describe the generalized Lie $n$-higher derivations. It is shown that, under some mild conditions, each component $G_r$ of a nonlinear generalized Lie $n$-higher derivation $\{G_r\}_{r\in N}$ of the triangular algebra $\mathcal{U}$ can be expressed as the sum of an additive generalized higher derivation and a nonlinear mapping vanishing on all ($n-1$)-th commutators on $\mathcal{U}$.

Author contributions

He Yuan: Conceptualization, Writing-original draft; Qian Zhang: Formal analysis, Writing-original draft; Zhendi Gu: Editing, Writing-original draft. All the contributors have perused and consented to the publishable draft of the manuscript.

Acknowledgments

This study was supported by Jilin Science and Technology Department (No. YDZJ202201ZYTS622).

Conflict of interest

The authors declare that they have no conflicts of interest.