Non-Global Lie Higher Derivations on Triangular Algebras Without Assuming Unity

Abstract

This work establishes a unified structural theory for non-global Lie higher derivations on triangular algebras $\mathcal{T}$, without assuming the existence of a unit element. The primary contribution is the introduction of extreme non-global Lie higher derivations and the proof that every non-global Lie higher derivation on $\mathcal{T}$ admits a unique decomposition into three components: a higher derivation, an extreme non-global Lie higher derivation, and a central map vanishing on all commutators $[x,y]$, where $x,y\in \mathcal{T}$ satisfy $xy=0$. This general framework is then explicitly applied to describe such derivations on two significant classes of algebras: upper triangular matrix algebras over faithful algebras and over semiprime algebras. By encompassing both unital and non-unital cases within a single characterization, the theory developed here not only generalizes numerous earlier results but also substantially expands the scope of the existing research landscape.

1. Introduction

Let $\mathcal{R}$ be a commutative ring with identity and let $\mathcal{A}$ be an algebra over $\mathcal{R}$, with its center denoted by $\mathcal{C}\left(\mathcal{A}\right)$. For any $x,y\in \mathcal{A}$, their Lie product is defined as $[x,y]=xy-yx$.

We now introduce several key concepts concerning mapping families. Let $\mathbb{N}$ be the set of non-negative integers and consider a family $\mathrm{\Delta }={\delta }_{n}n\in \mathbb{N}$ of $\mathcal{R}$-linear mappings on $\mathcal{A}$ satisfying ${\delta }_0=id\mathcal{A}$. Then $\mathrm{\Delta }$ is said to be

(a)

A higher derivation if for all $x,y\in \mathcal{A}$ and every $n\in \mathbb{N}$,

$${\delta }_n\left(xy\right)=\sum _{i+j=n}{\delta }_i\left(x\right){\delta }_j\left(y\right);$$

(b)

A Lie higher derivation if for all $x,y\in \mathcal{A}$ and every $n\in \mathbb{N}$,

$${\delta }_n\left([x,y]\right)=\sum _{i+j=n}[{\delta }_i\left(x\right),{\delta }_j\left(y\right)]$$

(c)

A non-global Lie higher derivation if for all $x,y\in \mathcal{A}$ (with $xy=0$) and every $n\in \mathbb{N}$,

$${\delta }_n\left([x,y]\right)=\sum _{i+j=n}[{\delta }_i\left(x\right),{\delta }_j\left(y\right)]$$

Note that a non-global Lie higher derivation is also referred to as a Lie higher derivation at zero products. Clearly, when $n=1$, (non-global) Lie higher derivations and higher derivations reduce to (non-global) Lie derivations and ordinary derivations, respectively. The study of (non-global) Lie derivations is highly active in the context of potentially non-associative or non-commutative algebras; see, for example, [1,2,3,4]. It is straightforward to verify that every higher derivation is a Lie higher derivation. However, the converse does not generally hold. Higher derivations play a fundamental role in algebraic theory and remain a vibrant research topic, including over non-associative or non-commutative algebras [5,6,7,8,9,10,11,12]. Throughout the evolution of derivation theory, both (Lie) derivations and their higher-order counterparts have continually attracted considerable attention. A central contributor to this area is Bresar [13,14,15,16], whose research especially on Herstein’s conjectures related to Lie mappings was instrumental in shaping the theory of functional identities, as comprehensively presented in [17].

In the study of operator algebras and Lie theory, triangular algebras represent a significant class with canonical structure. Derivations and their higher analogs, particularly Lie higher derivations, serve as pivotal tools for uncovering structural properties of algebras. Previous investigations have predominantly relied on the assumption of unity, which constrains the applicability of the resulting theories. Consequently, a systematic study of the structure and classification of non-global Lie higher derivations on the more general class of non-unital triangular algebras constitutes an essential and yet underdeveloped direction of research.

Lie-type mapping on triangular algebras is a very important research object, which enriches Herstein’s Lie mapping conjecture to some extent. Since Cheung [18] began to study Lie derivations on triangular algebras, many scholars have studied the structures of many mappings related to Lie derivations on the basis that triangular algebras contain identity elements. After that, many scholars have extended Cheung’s result [18] to Lie higher derivations from different perspectives and extended the previous conclusions. Li and Shen’s results [8] also extended the results of Lie higher derivations on nest algebras studied by Qi and Hou [19]. Moafian and Ebrahimi Vishki [20] gave the structure of Lie higher derivations on triangular algebras by the entries of matrices and obtained a similar conclusion to that shown by Li and Shen [8]. At the same time, some other scholars have studied the structure of Lie derivations at zero products. Ji and Qi [21] studied the non-global Lie derivation and also generalized the conclusion of Cheung [18]. Lin [22] then extended it to the structure of the non-global Lie higher derivation. It should be noted that the unit elements of triangular algebra play an important role in the research process of the above articles [1,8,19,20,22,23,24].

At this point, we are concerned that many researchers have studied the structure of Lie-type maps on triangular algebras without assuming unity in recent years. On the basis of [1], Wang [4] describes the structure of Lie derivations on triangular algebras without assuming unity, which can be decomposed into the following form:

$$L=\delta +\sigma +\tau ,$$

where $\delta$ is a derivation of $\mathcal{T}$, $\sigma$ is an extreme Lie derivation, defined in Definition 2 of $\mathcal{T}$, and $\tau$ is a linear mapping into the ordinary center ${\mathcal{C}}_1\left(\mathcal{T}\right)$, defined in Part 2, of $\mathcal{T}$ and $\tau \left(\right[\mathcal{T},\mathcal{T}\left]\right)=0$. After this, Liang and Ren [25] studied the structure of Lie higher derivations on triangular algebras without assuming unity and obtained that each Lie higher derivation can be written as the sum of higher derivations, extreme Lie higher derivations, and central maps; this result generalizes the results of Wang [4], Cheung [1], and Li and Shen [8]. We note that Ji and Qi [21] considered the structure of non-global Lie derivations on unital triangular algebras. To our surprise, Behfar and Ghahramani [26] also studied non-global Lie derivations on triangular algebras without assuming unity and proved that every non-global Lie derivation L can be decomposed into the following form:

$$L=\delta +\sigma +\tau ,$$

where $\delta$ is a derivation of $\mathcal{T}$, $\sigma$ is an non-global extreme Lie derivation of $\mathcal{T}$, and $\tau$ is a linear mapping into the ordinary center ${\mathcal{C}}_1\left(\mathcal{T}\right)$, defined in Section 2, of $\mathcal{T}$ and $\tau \left(\right[x,y\left]\right)=0$, with $xy=0$$x,y\in \mathcal{T}$. So an interesting question is proposed:

Question 1.

How can the structural form of non-global Lie higher derivations on triangular algebra be completely characterized without assuming unity?

This is the main purpose of this paper. This is a very interesting problem, and the characterization of the problem not only comprises a new topic, but also generalizes some existing conclusions ([26] (Theorem 4.3) and [4,21,25]).

Inspired by the article [4,21,25,26], the focus of this article is to characterize the structural form of non-global Lie higher derivations on triangular algebras without assuming unity. We provide sufficient conditions for each non-global Lie higher derivation on a triangular algebra $\left[\begin{array}{cc}A& M\\ O& B\end{array}\right]$ without assuming unity has a new decomposition form. That is, the following holds.

Theorem 1.

Let $\mathcal{T}=\left[\begin{array}{cc}A& M\\ O& B\end{array}\right]$ be a triangular algebra without assuming unity, if the following statements hold true:

(i) 

$\mathcal{C}\left(\mathcal{A}\right)={\pi }_{\mathcal{A}}\left(\mathcal{C}\left(\mathcal{T}\right)\right)$ and $\mathcal{C}\left(\mathcal{B}\right)={\pi }_{\mathcal{B}}\left(\mathcal{C}\left(\mathcal{T}\right)\right)$;

(ii) 

M is strong faithful;

(iii) 

For any $m_0\in M$, we have that $A{m}_0=0$ if and only if $m_{0}B=0$.

Let a sequence $\mathbb{L}={\left\{L_n\right\}}_{n\in \mathbb{N}}$ of linear mappings $L_n:\mathcal{T}\to \mathcal{T}$ be a non-global Lie higher derivation. Then there exists a higher derivation $\mathrm{\Delta }={\left\{{\delta }_n\right\}}_{n\in \mathbb{N}}$ of the linear mapping ${\delta }_n:\mathcal{T}\to \mathcal{T}$, a non-global extreme Lie higher derivation $\sum ={\left\{{\sigma }_n\right\}}_{n\in \mathbb{N}}$ of the linear mapping ${\sigma }_n:\mathcal{T}\to \mathcal{T}$, and an $\mathcal{R}$-linear mapping ${\tau }_n:\mathcal{T}\to {\mathcal{C}}_1\left(\mathcal{T}\right)$ vanishing on commutators $[x,y]$ with $xy=0$ such that

$$L_n\left(x\right)={\delta }_n\left(x\right)+{\sigma }_n\left(x\right)+{\tau }_n\left(x\right)$$

for all $x\in \mathcal{T}$.

And then we apply this result to describe the non-global Lie higher derivations on upper triangular matrix algebras $T_n\left(\mathcal{A}\right)$ over a faithful algebra $\mathcal{A}$ (see Corollary 4) and a semiprime algebra $\mathcal{A}$ (see Corollary 5), respectively.

In this paper, we focus on the structure of non-global Lie higher derivations on triangular algebras without assuming unity. This study topic has two aspects. Firstly, we assume that the triangular algebra does not contain the identity element, so we know from the research technique that the identity element of the triangular algebra is not used in the whole proof process. At the same time, it should be noted that although we assume that the triangular algebra does not contain identity elements, after careful inspection, it is found that the proof process can also be applied to a triangular algebra with identity elements with only a slight change. Secondly, non-global Lie higher derivations are a kind of very important research object, which are similar to the form of higher derivations in analysis to a certain extent and also the general generalization form of Lie derivations. Based on this, under certain conditions, this paper proves that each non-global Lie higher derivation has a completely new decomposition form (see Theorem 2 or Theorem 1). This is the first feature of this article. In addition, the proof process in this paper is also suitable for the case of a triangular algebra containing identity elements. Only by modifying the proof process appropriately can the conclusions of Behfar and Ghahramani’s [26], Liang and Ren’s [25], and Lin’s [22] results be obtained, which is the second feature of this paper.

2. Triangular Algebras Without Assuming Unity

This section outlines foundational concepts for our study. We first introduce an extended definition of a bimodule, which serve as a cornerstone in the theory of triangular algebras. While standard faithful bimodules have been extensively studied [26] (2. Preliminaries and Tools), Wang [4] recently proposed a significant generalization of this concept, which we now present.

Definition 1.

Let A and B be algebras. A bimodule M is called

• Left strongly faithful if for any $a\in A$, the condition $aMB=0$ implies $a=0$;
• Right strongly faithful if for any $b\in B$, the condition $AMb=0$ implies $b=0$;
• Strongly faithful if it is both left and right strongly faithful.

Clearly, strong faithfulness implies faithfulness for any bimodule. When A and B possess identities, the stronger notions of left/right strong faithfulness reduce precisely to the conventional left/right faithfulness found in the literature [18,27]. The distinction between these concepts is non-trivial, as demonstrated by several faithful but non-strongly faithful bimodules explicitly constructed in [4,26].

Based on this, we give the concept of triangular algebra. For algebras A and B, and their $(A,B)$-bimodule M, we define an algebra

$$\mathcal{T}=\left[\begin{array}{cc}A& M\\ 0& B\end{array}\right]=\left\{\left[\begin{array}{cc}a& m\\ 0& b\end{array}\right]|\forall a\in A,m\in M,b\in B\right\}$$

(1)

with the help of the usual matrix addition and multiplication operations. We observe the following elementary property: the triangular algebra $\mathcal{T}=\left[\begin{array}{cc}A& M\\ O& B\end{array}\right]$ admits a unit if and only if A and B are unital, which constitutes a necessary and sufficient condition. If algebra $\mathcal{T}$ contains unit elements, it evolves into a triangular algebra studied by many scholars [1,8,19,20,22].

Starting from this point, we assume that triangular algebras $\mathcal{T}$ do not contain identity elements. At this point, it should be noted that triangular algebra $\mathcal{T}$ without assuming unity still has the form of Equation (1). Using the standard representation (1) of a triangular algebra $\mathcal{T}$, we define the canonical $\mathcal{R}$-linear projections ${\pi }_A:\mathcal{T}\to A$ and ${\pi }_B:\mathcal{T}\to B$ by

$${\pi }_A:\left[\begin{array}{cc}a& m\\ 0& b\end{array}\right]⟼a\mathrm{and}{\pi }_B:\left[\begin{array}{cc}a& m\\ 0& b\end{array}\right]⟼b.$$

With the help of [4] (Proposition 2.1), the center $\mathcal{C}\left(\mathcal{T}\right)$ of algebra $\mathcal{T}$ is characterized as follows:

$$\mathcal{C}\left(\mathcal{T}\right)=\left\{\left[\begin{array}{cc}a& m_0\\ 0& b\end{array}\right]|am=mb,\forall m\in M\mathrm{and}A{m}_0=0=m_{0}B\right\}={\mathcal{C}}_1\left(\mathcal{T}\right)\oplus {\mathcal{C}}_2\left(\mathcal{T}\right)$$

(2)

where

$${\mathcal{C}}_1\left(\mathcal{T}\right)=\left\{\left[\begin{array}{cc}a& 0\\ 0& b\end{array}\right]|am=mb,\forall m\in M\right\}\mathrm{and}{\mathcal{C}}_2\left(\mathcal{T}\right)=\left\{\left[\begin{array}{cc}0& m_0\\ 0& 0\end{array}\right]|A{m}_0=0=m_{0}B,m_0\in M\right\}$$

are called the ordinary center and extreme center of $\mathcal{T}$ [4] (Proposition 2.1), respectively. Moreover, ${\pi }_A\left(\mathcal{C}\left(\mathcal{T}\right)\right)={\pi }_A\left({\mathcal{C}}_1\left(\mathcal{T}\right)\right)\subseteq \mathcal{C}\left(A\right)$ and ${\pi }_B\left(\mathcal{C}\left(\mathcal{T}\right)\right)={\pi }_B\left({\mathcal{C}}_1\left(\mathcal{T}\right)\right)\subseteq \mathcal{C}\left(B\right)$. Notice that if the algebra $\mathcal{T}$ has an identity element, then ${\mathcal{C}}_2\left(\mathcal{T}\right)=0$ and $\mathcal{C}\left(\mathcal{T}\right)={\mathcal{C}}_1\left(\mathcal{T}\right)$. The existence of a unit in a triangular algebra is crucial for isolating the bimodule M in central characterizations. The operation $am$ simplifies to m when $a=1$, a reduction impossible without a unit. Consequently, the inability to separate m from such products necessitates the introduction of an extreme center for a complete description.

Observe that ${\pi }_A\left(\mathcal{C}\left(\mathcal{T}\right)\right)$ and ${\pi }_B\left(\mathcal{C}\left(\mathcal{T}\right)\right)$ form subalgebras of $\mathcal{C}\left(A\right)$ and $\mathcal{C}\left(B\right)$, respectively. Moreover, there exists a unique canonical isomorphism $\eta :{\pi }_A\left(\mathcal{C}\left(\mathcal{T}\right)\right)\to {\pi }_B\left(\mathcal{C}\left(\mathcal{T}\right)\right)$ satisfying $am=m\eta \left(a\right)$ for all $a\in {\pi }_A\left(\mathcal{C}\left(\mathcal{T}\right)\right)$ and $m\in M$.

Based on the above definition of a center, we introduce a new concept, extreme non-global Lie higher derivation, which extends the definition of extreme Lie derivation at zero products in the article [26] (Definition 4.2). Extreme non-global Lie higher derivations provide a critical link for investigating the connection between non-global Lie higher derivations and the extreme center. Specifically, the vanishing of the extreme center forces any such derivation to reduce to the zero map.

Definition 2.

Let $\mathcal{T}=\left[\begin{array}{cc}A& M\\ O& B\end{array}\right]$ be a triangular algebra. A non-global Lie higher derivation $\sum ={\left\{{\sigma }_n\right\}}_{n\in \mathbb{N}}$ of $\mathcal{T}$ is said to be an extreme non-global Lie higher derivation if there exist a higher derivation $\mathrm{\Delta }={\left\{{\delta }_n\right\}}_{n\in \mathbb{N}}$ satisfy the following relationships:

(a) 

${\sigma }_n$ is a nonzero mapping on $\mathcal{T}$ and ${\sigma }_0=I{d}_{\mathcal{T}}$ is an identity mapping on $\mathcal{T}$;

(b) 

${\sigma }_n\left(xy\right)-{\sum }_{i+j=n}({\delta }_i\left(x\right){\sigma }_j\left(y\right)+{\sigma }_i\left(x\right){\delta }_j\left(y\right))\in {\mathcal{C}}_2\left(\mathcal{T}\right)$;

(c) 

${\sigma }_n\left([x,y]\right)={\sum }_{i+j=n}([{\delta }_i\left(x\right),{\sigma }_j\left(y\right)]+[{\sigma }_i\left(x\right),{\delta }_j\left(y\right)])$ with $xy=0$.

It should be noted that if algebra A contains a unit element, each extreme non-global Lie higher derivation degenerates to zero. During this research, we were surprised to find that there are nonzero extreme non-global Lie higher derivations in many algebras, such as [1] (Example 8) and [4] (Example 2.3).

The content studied in references [8,28,29] indeed concerns the structure of Lie-type derivations on triangular algebras. However, they all rely on the unit element of the triangular algebra. Comparing these studies [8,28,29] on Lie higher derivations, we characterize the structure of non-global Lie higher derivations on triangular algebras without assuming that triangular algebras contain unit elements and obtain a new decomposition structure (see Theorem 2) of non-global Lie higher derivation. From the proof of Theorem 2, it can be seen that this structure generalizes Behfar and Ghahramani’s results [26] (Theorem 4.4). In the research process, we just do not assume that there is a unit element. When this assumption is removed, the triangular algebra will evolve into a triangular algebra with a unit element in Section 3. Therefore, the strong faithful bimodule used in Theorem 2 will evolve into a faithful bimodule, and the extreme center will evolve into zero. Therefore, Theorem 2 generalizes some existing conclusions.

3. Non-Global Lie Higher Derivations

As a starting point for characterizing non-global Lie higher derivations, we present a key lemma. It offers an equivalent characterization for higher derivations on the triangular algebra $\mathcal{T}=\left[\begin{array}{cc}A& M\\ O& B\end{array}\right]$, dispensing with the assumption of unity and only requiring M to be strongly faithful.

Lemma 1

([25] (Theorem 3.1)). Let $\mathcal{T}=\left[\begin{array}{cc}A& M\\ O& B\end{array}\right]$ with M strong faithful. A sequence $\mathrm{\Delta }={\left\{{\delta }_n\right\}}_{n\in \mathbb{N}}$ of linear mappings ${\delta }_n:\mathcal{T}\to \mathcal{T}$ is a higher derivation associated with local actions if and only if for each $n\in \mathbb{N}$, ${\delta }_n$ can be written as

$${\delta }_n\left(\left[\begin{array}{cc}a& m\\ 0& b\end{array}\right]\right)=\left[\begin{array}{cc}{p}_A^{\left(n\right)}\left(a\right)& r_1^{\left(n\right)}\left(a\right)-r_2^{\left(n\right)}\left(b\right)+f^{\left(n\right)}\left(m\right)\\ 0& p_B^{\left(n\right)}\left(b\right)\end{array}\right],$$

where, for arbitrary $n\in \mathbb{N}$,

$$\begin{array}{cc}& p_A^{\left(n\right)}:A\to A,p_B^{\left(n\right)}:B\to B;\hfill \\ & r_1^{\left(n\right)}:A\to M,r_2^{\left(n\right)}:B\to M,f^{\left(n\right)}:M\to M\hfill \end{array}$$

are all $\mathcal{R}$-linear mappings satisfying the following conditions:

(i) 

$f^{\left(n\right)}\left(am\right)={\sum }_{i+j=n}{p}_A^{\left(i\right)}\left(a\right)f^{\left(j\right)}\left(m\right)$;

(ii) 

$f^{\left(n\right)}\left(mb\right)={\sum }_{i+j=n}{f}^{\left(i\right)}\left(m\right)p_B^{\left(j\right)}\left(b\right)$;

(iii) 

${\sum }_{i+j=n}{p}_A^{\left(i\right)}\left(a\right)r_2^{\left(j\right)}\left(b\right)={\sum }_{i+j=n}{r}_1^{\left(i\right)}\left(a\right)p_B^{\left(j\right)}\left(b\right)$,

(iv) 

$r_1^{\left(n\right)}\left(a{a}^{\prime }\right)={\sum }_{i+j=n}{p}_A^{\left(i\right)}\left(a\right)r_1^{\left(j\right)}\left(a^{\prime }\right)$, and $r_2^{\left(n\right)}\left(b{b}^{\prime }\right)={\sum }_{i+j=n}{r}_2^{\left(i\right)}\left(b\right)p_B^{\left(j\right)}\left(b^{\prime }\right)$

for all $a,a^{\prime }\in A$ and $b,b^{\prime }\in B$.

We now give the main result of this paper.

Theorem 2.

Let $\mathcal{T}=\left[\begin{array}{cc}A& M\\ O& B\end{array}\right]$ be a triangular algebra, if the following statements hold true:

(i) 

$\mathcal{C}\left(\mathcal{A}\right)={\pi }_{\mathcal{A}}\left(\mathcal{C}\left(\mathcal{T}\right)\right)$ and $\mathcal{C}\left(\mathcal{B}\right)={\pi }_{\mathcal{B}}\left(\mathcal{C}\left(\mathcal{T}\right)\right)$;

(ii) 

M is strong faithful;

(iii) 

For any $m_0\in M$, we have that $A{m}_0=0$ if and only if $m_{0}B=0$.

Let a sequence $\mathbb{L}={\left\{L_n\right\}}_{n\in \mathbb{N}}$ of linear mappings $L_n:\mathcal{T}\to \mathcal{T}$ be a non-global Lie higher derivation. Then there exists a higher derivation $\mathrm{\Delta }={\left\{{\delta }_n\right\}}_{n\in \mathbb{N}}$ of the linear mapping ${\delta }_n:\mathcal{T}\to \mathcal{T}$, an extreme non-global Lie higher derivation $\sum ={\left\{{\sigma }_n\right\}}_{n\in \mathbb{N}}$ of the linear mapping ${\sigma }_n:\mathcal{T}\to \mathcal{T}$, and an $\mathcal{R}$-linear mapping ${\tau }_n:\mathcal{T}\to {\mathcal{C}}_1\left(\mathcal{T}\right)$ vanishing on commutators $[x,y]$ with $xy=0$ such that

$$L_n\left(x\right)={\delta }_n\left(x\right)+{\sigma }_n\left(x\right)+{\tau }_n\left(x\right)$$

for all $x\in \mathcal{T}$.

Proof. 

Suppose that a sequence $\mathbb{L}={\left\{L_n\right\}}_{n\in \mathbb{N}}$ of linear mappings $L_n:\mathcal{T}\to \mathcal{T}$ is a non-global Lie higher derivation. Write $L_n$ as

$$L_n\left(\left[\begin{array}{cc}a& m\\ 0& b\end{array}\right]\right)=\left[\begin{array}{cc}{\alpha }_1^{\left(n\right)}\left(a\right)+{\beta }_1^{\left(n\right)}\left(b\right)+{\gamma }_1^{\left(n\right)}\left(m\right)& {\alpha }_2^{\left(n\right)}\left(a\right)+{\beta }_2^{\left(n\right)}\left(b\right)+{\gamma }_2^{\left(n\right)}\left(m\right)\\ 0& {\alpha }_3^{\left(n\right)}\left(a\right)+{\beta }_3^{\left(n\right)}\left(b\right)+{\gamma }_3^{\left(n\right)}\left(m\right)\end{array}\right]$$

for all $a\in A,b\in B$ and $m\in M$, where $L_0=I{d}_{\mathcal{T}}$ is an identity map on algebra $\mathcal{T}$, ${\alpha }_1^{\left(n\right)},{\beta }_1^{\left(n\right)},{\gamma }_1^{\left(n\right)}$ are linear mappings from $A,M,B$ to A, respectively; ${\alpha }_2^{\left(n\right)},{\beta }_2^{\left(n\right)},{\gamma }_2^{\left(n\right)}$ are linear mappings from $A,M,B$ to M, respectively; and ${\alpha }_3^{\left(n\right)},{\beta }_3^{\left(n\right)},{\gamma }_3^{\left(n\right)}$ are linear mappings from $A,M,B$ to B, respectively.

When $n=1$, the mapping $L_1$ is equal to mapping $\mathrm{\Delta }$ in [26] (Theorem 4.4). According to $\left[\begin{array}{cc}0& 0\\ 0& b\end{array}\right]\left[\begin{array}{cc}a& 0\\ 0& 0\end{array}\right]=\left[\begin{array}{cc}a& 0\\ 0& 0\end{array}\right]\left[\begin{array}{cc}0& 0\\ 0& b\end{array}\right]$, we have

$$\begin{array}{cc}\hfill 0& =L_n\left(\left[\left[\begin{array}{cc}0& 0\\ 0& b\end{array}\right],\left[\begin{array}{cc}a& 0\\ 0& 0\end{array}\right]\right]\right)\hfill \\ & =\sum _{i+j=n}\left[L_i\left(\left[\begin{array}{cc}0& 0\\ 0& b\end{array}\right]\right),L_j\left(\left[\begin{array}{cc}a& 0\\ 0& 0\end{array}\right]\right)\right]=\sum _{i+j=n}\left[\left[\begin{array}{cc}{\beta }_1^{\left(i\right)}\left(b\right)& {\beta }_2^{\left(i\right)}\left(b\right)\\ 0& {\beta }_3^{\left(i\right)}\left(b\right)\end{array}\right],\left[\begin{array}{cc}{\alpha }_1^{\left(j\right)}\left(a\right)& {\alpha }_2^{\left(j\right)}\left(a\right)\\ 0& {\alpha }_3^{\left(j\right)}\left(a\right)\end{array}\right]\right]\hfill \\ & =\sum _{i+j=n}\left[\begin{array}{cc}{\beta }_1^{\left(i\right)}\left(b\right){\alpha }_1^{\left(j\right)}\left(a\right)-{\alpha }_1^{\left(j\right)}\left(a\right){\beta }_1^{\left(i\right)}\left(b\right)& {\beta }_1^{\left(i\right)}\left(b\right){\alpha }_2^{\left(j\right)}\left(a\right)+{\beta }_2^{\left(i\right)}\left(b\right){\alpha }_3^{\left(j\right)}\left(a\right)-{\alpha }_1^{\left(j\right)}\left(a\right){\beta }_2^{\left(i\right)}\left(b\right)-{\alpha }_2^{\left(j\right)}\left(a\right){\beta }_3^{\left(i\right)}\left(b\right)\\ 0& {\beta }_3^{\left(i\right)}\left(b\right){\alpha }_3^{\left(j\right)}\left(a\right)-{\alpha }_3^{\left(j\right)}\left(a\right){\beta }_3^{\left(i\right)}\left(b\right)\end{array}\right]\hfill \end{array}$$

for all $a\in A,b\in B$. Thus we arrive at

$$\begin{array}{cc}\hfill & \sum _{i+j=n}({\beta }_1^{\left(i\right)}\left(b\right){\alpha }_1^{\left(j\right)}\left(a\right)-{\alpha }_1^{\left(j\right)}\left(a\right){\beta }_1^{\left(i\right)}\left(b\right))=0,\hfill \\ \hfill & \sum _{i+j=n}({\beta }_3^{\left(i\right)}\left(b\right){\alpha }_3^{\left(j\right)}\left(a\right)-{\alpha }_3^{\left(j\right)}\left(a\right){\beta }_3^{\left(i\right)}\left(b\right))=0,\hfill \\ \hfill & \mathrm{and}\hfill \\ \hfill & \sum _{i+j=n}({\beta }_1^{\left(i\right)}\left(b\right){\alpha }_2^{\left(j\right)}\left(a\right)+{\beta }_2^{\left(i\right)}\left(b\right){\alpha }_3^{\left(j\right)}\left(a\right)-{\alpha }_1^{\left(j\right)}\left(a\right){\beta }_2^{\left(i\right)}\left(b\right)-{\alpha }_2^{\left(j\right)}\left(a\right){\beta }_3^{\left(i\right)}\left(b\right))=0\hfill \end{array}$$

(3)

for all $a\in A,b\in B$.

Let us check

$${\alpha }_3^{\left(n\right)}\left(b\right)\in \mathcal{C}\left(A\right)\mathrm{and}{\beta }_1^{\left(n\right)}\left(a\right)\in \mathcal{C}\left(B\right)$$

(4)

by complete induction on n for all $a\in A,b\in B$. When $n=1$, due to the proof of [26] (Theorem 4.4), it is clear that

$${\alpha }_3^{\left(1\right)}\left(b\right)\in \mathcal{C}\left(A\right)\mathrm{and}{\beta }_1^{\left(1\right)}\left(a\right)\in \mathcal{C}\left(B\right)$$

for all $a\in A,b\in B$.

Suppose that

$${\alpha }_3^{\left(i\right)}\left(b\right)\in \mathcal{C}\left(A\right)\mathrm{and}{\beta }_1^{\left(j\right)}\left(a\right)\in \mathcal{C}\left(B\right)$$

for all $1\le i,j\le n-1$. With the help of (3), we therefore get

$$0=\sum _{i+j=n}({\beta }_1^{\left(i\right)}\left(b\right){\alpha }_1^{\left(j\right)}\left(a\right)-{\alpha }_1^{\left(j\right)}\left(a\right){\beta }_1^{\left(i\right)}\left(b\right))={\beta }_1^{\left(n\right)}\left(b\right)a-a{\beta }_1^{\left(n\right)}\left(b\right)$$

and

$$0=\sum _{i+j=n}({\alpha }_1^{\left(i\right)}\left(a\right){\beta }_1^{\left(j\right)}\left(b\right)-{\beta }_1^{\left(j\right)}\left(b\right){\alpha }_1^{\left(i\right)}\left(a\right))=b{\alpha }_3^{\left(n\right)}\left(a\right)-{\alpha }_3^{\left(n\right)}\left(a\right)b$$

for all $a\in A,b\in B$. And then ${\beta }_1^{\left(n\right)}\left(b\right)\in \mathcal{C}\left(A\right)$ and ${\alpha }_3^{\left(n\right)}\left(a\right)\in \mathcal{C}\left(B\right)$ for all $a\in A,b\in B$. Taking into account (4), the equation in (3) can be rewritten as

$$\sum _{i+j=n}({\alpha }_2^{\left(j\right)}\left(a\right)(\eta ({\beta }_1^{\left(i\right)}\left(b\right))-{\beta }_3^{\left(i\right)}\left(b\right))-(-{\eta }^{-1}({\alpha }_3^{\left(j\right)}\left(a\right))+{\alpha }_1^{\left(j\right)}\left(a\right)){\beta }_2^{\left(i\right)}\left(b\right))=0$$

(5)

for all $a\in A,b\in B$. Let us define $p_A^{\left(j\right)}\left(a\right)={\alpha }_1^{\left(j\right)}\left(a\right)-{\eta }^{-1}({\alpha }_3^{\left(j\right)}\left(a\right))$ and $p_B^{\left(i\right)}\left(b\right)={\beta }_3^{\left(i\right)}\left(b\right)-\eta ({\beta }_1^{\left(i\right)}\left(b\right))$ for all $1\le i,j\le n$. Thus Equation (5) can be rewritten as

$$\sum _{i+j=n}({\alpha }_2^{\left(j\right)}\left(a\right)p_B^{\left(i\right)}\left(b\right)-p_A^{\left(j\right)}\left(a\right){\beta }_2^{\left(i\right)}\left(b\right))=0.$$

(6)

For any $a\in A,m\in M$, we have $\left[\begin{array}{cc}0& m\\ 0& 0\end{array}\right]\left[\begin{array}{cc}a& 0\\ 0& 0\end{array}\right]=0$ and $\left[\begin{array}{cc}a& 0\\ 0& 0\end{array}\right]\left[\begin{array}{cc}0& m\\ 0& 0\end{array}\right]=\left[\begin{array}{cc}0& am\\ 0& 0\end{array}\right]$; this implies

$$\begin{array}{cc}& L_n\left(\left[\left[\begin{array}{cc}0& m\\ 0& 0\end{array}\right],\left[\begin{array}{cc}a& 0\\ 0& 0\end{array}\right]\right]\right)=\sum _{i+j=n}\left[L_i\left(\left[\begin{array}{cc}0& m\\ 0& 0\end{array}\right]\right),L_j\left(\left[\begin{array}{cc}a& 0\\ 0& 0\end{array}\right]\right)\right]\hfill \\ & =\sum _{i+j=n}\left[\left[\begin{array}{cc}{\gamma }_1^{\left(i\right)}\left(m\right)& {\gamma }_2^{\left(i\right)}\left(m\right)\\ 0& {\gamma }_3^{\left(i\right)}\left(m\right)\end{array}\right],\left[\begin{array}{cc}{\alpha }_1^{\left(j\right)}\left(a\right)& {\alpha }_2^{\left(j\right)}\left(a\right)\\ 0& {\alpha }_3^{\left(j\right)}\left(a\right)\end{array}\right]\right]\hfill \\ & =\sum _{i+j=n}\left[\begin{array}{cc}[{\gamma }_1^{\left(i\right)}\left(m\right),{\alpha }_1^{\left(j\right)}\left(a\right)]& U\\ 0& [{\gamma }_3^{\left(i\right)}\left(m\right),{\alpha }_3^{\left(j\right)}\left(a\right)]\end{array}\right],\hfill \end{array}$$

where

$$U={\gamma }_1^{\left(i\right)}\left(m\right){\alpha }_2^{\left(j\right)}\left(a\right)+{\gamma }_2^{\left(i\right)}\left(m\right){\alpha }_3^{\left(j\right)}\left(a\right)-{\alpha }_1^{\left(j\right)}\left(a\right){\gamma }_2^{\left(i\right)}\left(m\right)-{\alpha }_2^{\left(j\right)}\left(a\right){\gamma }_3^{\left(i\right)}\left(m\right)$$

for all $a\in A$ and $m\in M$.

On the other hand,

$$L_n\left(\left[\left[\begin{array}{cc}0& m\\ 0& 0\end{array}\right],\left[\begin{array}{cc}a& 0\\ 0& 0\end{array}\right]\right]\right)=L_n\left(\left[\begin{array}{cc}0& -am\\ 0& 0\end{array}\right]\right)=\left[\begin{array}{cc}{\gamma }_1^{\left(n\right)}(-am)& {\gamma }_2^{\left(n\right)}(-am)\\ 0& {\gamma }_3^{\left(n\right)}(-am)\end{array}\right]$$

for all $a\in A,b\in B$ and $m\in M$.

On comparing the above two relations, we see that

$${\gamma }_1^{\left(n\right)}(-am)=\sum _{i+j=n}[{\gamma }_1^{\left(i\right)}\left(m\right),{\alpha }_1^{\left(j\right)}\left(a\right)],{\gamma }_3^{\left(n\right)}(-am)=\sum _{i+j=n}[{\gamma }_3^{\left(i\right)}\left(m\right),{\alpha }_3^{\left(j\right)}\left(a\right)]$$

and

$$\begin{array}{cc}\hfill {\gamma }_2^{\left(n\right)}(-am)& =\sum _{i+j=n}({\gamma }_1^{\left(i\right)}\left(m\right){\alpha }_2^{\left(j\right)}\left(a\right)+{\gamma }_2^{\left(i\right)}\left(m\right){\alpha }_3^{\left(j\right)}\left(a\right)\hfill \\ \hfill & -{\alpha }_1^{\left(j\right)}\left(a\right){\gamma }_2^{\left(i\right)}\left(m\right)-{\alpha }_2^{\left(j\right)}\left(a\right){\gamma }_3^{\left(i\right)}\left(m\right))\hfill \\ \hfill & =\sum _{i+j=n}(-{\alpha }_1^{\left(i\right)}\left(a\right)+{\eta }^{-1}({\alpha }_3^{\left(j\right)}\left(a\right)){\gamma }_2^{\left(i\right)}\left(m\right)\hfill \\ \hfill & +{\gamma }_1^{\left(i\right)}\left(a\right){\alpha }_2^{\left(j\right)}\left(a\right)-{\alpha }_2^{\left(j\right)}\left(a\right){\gamma }_3^{\left(i\right)}\left(m\right)\hfill \end{array}$$

(7)

for all $a\in A$, $m\in M$.

Using the same methods, for any $a\in A,m\in M$, we have $\left[\begin{array}{cc}0& 0\\ 0& b\end{array}\right]\left[\begin{array}{cc}0& m\\ 0& 0\end{array}\right]=0$ and $\left[\begin{array}{cc}0& m\\ 0& 0\end{array}\right]\left[\begin{array}{cc}0& 0\\ 0& b\end{array}\right]=\left[\begin{array}{cc}0& mb\\ 0& 0\end{array}\right]$; this implies

$$\begin{array}{cc}& L_n\left(\left[\left[\begin{array}{cc}0& 0\\ 0& b\end{array}\right],\left[\begin{array}{cc}0& m\\ 0& 0\end{array}\right]\right]\right)=\sum _{i+j=n}\left[L_i\left(\left[\begin{array}{cc}0& 0\\ 0& b\end{array}\right]\right),L_i\left(\left[\begin{array}{cc}0& m\\ 0& 0\end{array}\right]\right)\right]\hfill \\ & =\sum _{i+j=n}\left[\left[\begin{array}{cc}{\beta }_1^{\left(i\right)}\left(b\right)& {\beta }_2^{\left(i\right)}\left(b\right)\\ 0& {\beta }_3^{\left(i\right)}\left(b\right)\end{array}\right],\left[\begin{array}{cc}{\gamma }_1^{\left(j\right)}\left(m\right)& {\gamma }_2^{\left(j\right)}\left(m\right)\\ 0& {\gamma }_3^{\left(j\right)}\left(m\right)\end{array}\right]\right]\hfill \\ & =\sum _{i+j=n}\left[\begin{array}{cc}[{\beta }_1^{\left(i\right)}\left(b\right),{\gamma }_1^{\left(j\right)}\left(m\right)]& U\\ 0& [{\beta }_3^{\left(i\right)}\left(b\right),{\gamma }_3^{\left(j\right)}\left(m\right)]\end{array}\right],\hfill \end{array}$$

where

$$U={\beta }_1^{\left(i\right)}\left(b\right){\gamma }_2^{\left(j\right)}\left(m\right)+{\beta }_2^{\left(i\right)}\left(b\right){\gamma }_3^{\left(j\right)}\left(m\right)-{\gamma }_1^{\left(j\right)}\left(m\right){\beta }_2^{\left(i\right)}\left(b\right)-{\gamma }_2^{\left(j\right)}\left(m\right){\beta }_3^{\left(i\right)}\left(b\right)$$

for all $a\in A$ and $m\in M$.

On the other hand,

$$L_n\left(\left[\left[\begin{array}{cc}0& 0\\ 0& b\end{array}\right],\left[\begin{array}{cc}0& m\\ 0& 0\end{array}\right]\right]\right)=L_n\left(\left[\begin{array}{cc}0& -mb\\ 0& 0\end{array}\right]\right)=\left[\begin{array}{cc}{\gamma }_1^{\left(n\right)}(-mb)& {\gamma }_2^{\left(n\right)}(-mb)\\ 0& {\gamma }_3^{\left(n\right)}(-mb)\end{array}\right]$$

for all $a\in A,b\in B$ and $m\in M$.

On comparing the above two relations, we see that

$${\gamma }_1^{\left(n\right)}(-mb)=\sum _{i+j=n}[{\beta }_1^{\left(i\right)}\left(b\right),{\gamma }_1^{\left(j\right)}\left(m\right)],{\gamma }_3^{\left(n\right)}(-mb)=\sum _{i+j=n}[{\beta }_3^{\left(i\right)}\left(b\right),{\gamma }_3^{\left(j\right)}\left(m\right)]$$

(8)

and

$$\begin{array}{cc}\hfill {\gamma }_2^{\left(n\right)}(-mb)& =\sum _{i+j=n}({\beta }_1^{\left(i\right)}\left(b\right){\gamma }_2^{\left(j\right)}\left(m\right)+{\beta }_2^{\left(i\right)}\left(b\right){\gamma }_3^{\left(j\right)}\left(m\right)\hfill \\ \hfill & -{\gamma }_1^{\left(j\right)}\left(m\right){\beta }_2^{\left(i\right)}\left(b\right)-{\gamma }_2^{\left(j\right)}\left(m\right){\beta }_3^{\left(i\right)}\left(b\right))\hfill \end{array}$$

(9)

for all $b\in B$ and $m\in M$.

For arbitrary $m^{\prime },m\in M$, we have the relation $\left[\begin{array}{cc}0& m\\ 0& 0\end{array}\right]\left[\begin{array}{cc}0& m^{\prime }\\ 0& 0\end{array}\right]=0.$ And then

$$\begin{array}{cc}\hfill 0& =L_n\left(\left[\left[\begin{array}{cc}0& m\\ 0& 0\end{array}\right],\left[\begin{array}{cc}0& m^{\prime }\\ 0& 0\end{array}\right]\right]\right)\hfill \\ & =\sum _{i+j=n}\left[L_i\left(\left[\begin{array}{cc}0& m\\ 0& 0\end{array}\right]\right),L_j\left(\left[\begin{array}{cc}0& m^{\prime }\\ 0& 0\end{array}\right]\right)\right]\hfill \\ & =\sum _{i+j=n}\left[\left[\begin{array}{cc}{\gamma }_1^{\left(i\right)}\left(m\right)& {\gamma }_2^{\left(i\right)}\left(m\right)\\ 0& {\gamma }_3^{\left(i\right)}\left(m\right)\end{array}\right],\left[\begin{array}{cc}{\gamma }_1^{\left(j\right)}\left(m^{\prime }\right)& {\gamma }_2^{\left(j\right)}\left(m^{\prime }\right)\\ 0& {\gamma }_3^{\left(j\right)}\left(m^{\prime }\right)\end{array}\right]\right]\hfill \\ & =\sum _{i+j=n}\left[\begin{array}{cc}[{\gamma }_1^{\left(i\right)}\left(m\right),{\gamma }_1^{\left(j\right)}\left(m^{\prime }\right)]& {\gamma }_1^{\left(i\right)}\left(m\right){\gamma }_2^{\left(j\right)}\left(m^{\prime }\right)+{\gamma }_2^{\left(i\right)}\left(m\right){\gamma }_3^{\left(j\right)}\left(m^{\prime }\right)-{\gamma }_1^{\left(j\right)}\left(m^{\prime }\right){\gamma }_2^{\left(i\right)}\left(m\right)-{\gamma }_2^{\left(j\right)}\left(m^{\prime }\right){\gamma }_3^{\left(i\right)}\left(m\right)\\ 0& [{\gamma }_3^{\left(i\right)}\left(m\right),{\gamma }_3^{\left(j\right)}\left(m^{\prime }\right)]\end{array}\right].\hfill \end{array}$$

It follows that

$$\begin{array}{cc}\hfill & \sum _{i+j=n}({\gamma }_1^{\left(i\right)}\left(m\right){\gamma }_2^{\left(j\right)}\left(m^{\prime }\right)+{\gamma }_2^{\left(i\right)}\left(m\right){\gamma }_3^{\left(j\right)}\left(m^{\prime }\right)-{\gamma }_1^{\left(j\right)}\left(m^{\prime }\right){\gamma }_2^{\left(i\right)}\left(m\right)-{\gamma }_2^{\left(j\right)}\left(m^{\prime }\right){\gamma }_3^{\left(i\right)}\left(m\right))\hfill \\ \hfill =& \sum _{i+j=n,j\ne 0\mathrm{or}i\ne 0}({\gamma }_1^{\left(i\right)}\left(m\right){\gamma }_2^{\left(j\right)}\left(m^{\prime }\right)+{\gamma }_2^{\left(i\right)}\left(m\right){\gamma }_3^{\left(j\right)}\left(m^{\prime }\right)-{\gamma }_1^{\left(j\right)}\left(m^{\prime }\right){\gamma }_2^{\left(i\right)}\left(m\right)-{\gamma }_2^{\left(j\right)}\left(m^{\prime }\right){\gamma }_3^{\left(i\right)}\left(m\right))\hfill \\ \hfill & +{\gamma }_1^{\left(n\right)}\left(m\right)m^{\prime }-m^{\prime }{\gamma }_3^{\left(n\right)}\left(m\right)-{\gamma }_1^{\left(n\right)}\left(m^{\prime }\right)m+m{\gamma }_3^{\left(n\right)}\left(m^{\prime }\right)\hfill \\ \hfill =0\end{array}$$

(10)

for all $m,m^{\prime }\in M$.

Our focus is on proving the following conclusion:

$${\gamma }_1^{\left(n\right)}\left(mb\right)=0,{\gamma }_3^{\left(n\right)}\left(am\right)=0,\mathrm{and}{\gamma }_1^{\left(n\right)}\left(m\right)\oplus {\gamma }_3^{\left(n\right)}\left(m\right)\in {\mathcal{C}}_1\left(\mathcal{T}\right)$$

for all $a\in A,m\in M,b\in B$. We are going to prove this by mathematical induction for n.

By the proof of [26] (Theorem 3.2), we know that

$${\gamma }_1^{\left(1\right)}\left(mb\right)=0,{\gamma }_3^{\left(1\right)}\left(am\right)=0,\mathrm{and}{\gamma }_1^{\left(1\right)}\left(m\right)\oplus {\gamma }_3^{\left(1\right)}\left(m\right)\in {\mathcal{C}}_1\left(\mathcal{T}\right)$$

whenever $n=1$ for all $a\in A,m\in M,b\in B$. Suppose that

$${\gamma }_1^{\left(x_1\right)}\left(mb\right)=0,{\gamma }_3^{\left(x_2\right)}\left(am\right)=0,\mathrm{and}{\gamma }_1^{\left(x_3\right)}\left(m\right)\oplus {\gamma }_3^{\left(x_3\right)}\left(m\right)\in {\mathcal{C}}_1\left(\mathcal{T}\right)$$

(11)

whenever $1\le x_1,x_2,x_3\le n-1$ for all $a\in A,m\in M,b\in B$. In the process of deforming the equation, we use mathematical induction to obtain

$${\gamma }_1^{\left(n\right)}(-mb)=\sum _{i+j=n}[{\beta }_1^{\left(i\right)}\left(b\right),{\gamma }_1^{\left(j\right)}\left(m\right)]=\sum _{i+j=n,i\ne 0}[{\beta }_1^{\left(i\right)}\left(b\right),{\gamma }_1^{\left(j\right)}\left(m\right)]=0$$

(12)

and

$${\gamma }_3^{\left(n\right)}(-am)=\sum _{i+j=n}[{\gamma }_3^{\left(i\right)}\left(m\right),{\alpha }_3^{\left(j\right)}\left(a\right)]=\sum _{i+j=n,j\ne 0}[{\gamma }_3^{\left(i\right)}\left(m\right),{\alpha }_3^{\left(j\right)}\left(a\right)]=0$$

(13)

via ${\gamma }_1^{\left(j\right)}\left(m\right)\in \mathcal{C}\left(A\right)$ and ${\gamma }_3^{\left(i\right)}\left(m\right)\in \mathcal{C}\left(B\right)$ for $1\le i,j\le n-1$. Meanwhile, comparing with Equation (10) and using complete mathematical induction, we therefore arrive at

$${\gamma }_1^{\left(n\right)}\left(m\right)m^{\prime }-m^{\prime }{\gamma }_3^{\left(n\right)}\left(m\right)-{\gamma }_1^{\left(n\right)}\left(m^{\prime }\right)m+m{\gamma }_3^{\left(n\right)}\left(m^{\prime }\right)=0$$

for all $m,m^{\prime }\in M$. Now substituting $a{m}^{\prime }b$ in the above equation instead of $m^{\prime }$ and using (12), (13) together with the induction hypothesis (11),

$${\gamma }_1^{\left(n\right)}\left(m\right)a{m}^{\prime }b-a{m}^{\prime }b{\gamma }_3^{\left(n\right)}\left(m\right)=0$$

(14)

for all $a\in A,b\in B$ and $m,m^{\prime }\in M$. Also substituting $a^{\prime }a$ in the above equation instead of a we obtain

$${\gamma }_1^{\left(n\right)}\left(m\right)a{a}^{\prime }{m}^{\prime }b-a{a}^{\prime }{m}^{\prime }b{\gamma }_3^{\left(n\right)}\left(m\right)=0$$

for all $a,a^{\prime }\in A,b\in B$ and $m,m^{\prime }\in M$. Now by multiplying $a^{\prime }$ with Equation (14) from the left we find

$$a^{\prime }{\gamma }_1^{\left(n\right)}\left(m\right)a{m}^{\prime }b-a^{\prime }a{m}^{\prime }b{\gamma }_3^{\left(n\right)}\left(m\right)=0.$$

By comparing these identities we have

$$({\gamma }_1^{\left(n\right)}\left(m\right)a^{\prime }-a^{\prime }{\gamma }_1^{\left(n\right)}\left(m\right))a{m}^{\prime }b=0$$

for all $a,a^{\prime }\in A,b\in B$ and $m,m^{\prime }\in M$. Given the arbitrariness of elements $a\in A$ and $b\in B$, we can obtain

$$({\gamma }_1^{\left(n\right)}\left(m\right)a^{\prime }-a^{\prime }{\gamma }_1^{\left(n\right)}\left(m\right))AMB=0$$

for all $a^{\prime }\in A$ and $m\in M$. By assumption this implies

$${\gamma }_1^{\left(n\right)}\left(m\right)a^{\prime }-a^{\prime }{\gamma }_1^{\left(n\right)}\left(m\right)=0$$

for all $a^{\prime }\in A,m\in M$. Therefore, ${\gamma }_1^{\left(n\right)}\left(m\right)\in \mathcal{C}\left(A\right)$ for all $m\in M$. Since ${\pi }_A\left(\mathcal{C}\left(\mathcal{T}\right)\right)=\mathcal{C}\left(A\right)$, there exist $\eta \left({\gamma }_1^{\left(n\right)}\left(m\right)\right)\in \mathcal{C}\left(B\right)$ such that

$${\gamma }_1^{\left(n\right)}\left(m\right)m^{\prime }=m^{\prime }\eta \left({\gamma }_1^{\left(n\right)}\left(m\right)\right)$$

for all $m^{\prime },m\in M$. It follows from equation ${\gamma }_1^{\left(n\right)}\left(m\right)a{m}^{\prime }b-a{m}^{\prime }b{\gamma }_3^{\left(n\right)}\left(m\right)=0$ that $a{m}^{\prime }b\eta \left({\gamma }_1^{\left(n\right)}\left(m\right)\right)-a{m}^{\prime }b{\gamma }_3^{\left(n\right)}\left(m\right)=0$ for all $a\in A,b\in B,m^{\prime },m\in M$. Therefore, $AMB(\eta \left({\gamma }_1^{\left(n\right)}\left(m\right)\right)-{\gamma }_3^{\left(n\right)}\left(m\right))=0$ for all $m\in M$. According to the strong faithfulness of bimodule M, we can get

$$\eta ({\gamma }_1^{\left(n\right)}\left(m\right))={\gamma }_3^{\left(n\right)}\left(m\right),$$

for all $m\in M$. Therefore,

$$\left[\begin{array}{cc}{\gamma }_1^{\left(i\right)}\left(m\right)& \\ 0& {\gamma }_3^{\left(i\right)}\left(m\right)\end{array}\right]\in {\mathcal{C}}_1\left(\mathcal{T}\right).$$

(15)

for arbitrary $1\le i\le n$. At this point, we prove that the induction hypothesis holds.

Corresponding to Equations (7), (9), and (15), we have

$$\begin{array}{cc}\hfill {\gamma }_2^{\left(n\right)}(-am)& =\sum _{i+j=n}({\gamma }_1^{\left(i\right)}\left(m\right){\alpha }_2^{\left(j\right)}\left(a\right)+{\gamma }_2^{\left(i\right)}\left(m\right){\alpha }_3^{\left(j\right)}\left(a\right)\hfill \\ & -{\alpha }_1^{\left(j\right)}\left(a\right){\gamma }_2^{\left(i\right)}\left(m\right)-{\alpha }_2^{\left(j\right)}\left(a\right){\gamma }_3^{\left(i\right)}\left(m\right))\hfill \\ & =\sum _{i+j=n}({\eta }^{-1}({\alpha }_3^{\left(j\right)}\left(a\right))-{\alpha }_1^{\left(j\right)}\left(a\right)){\gamma }_2^{\left(i\right)}\left(m\right)\hfill \\ & =-\sum _{i+j=n}{p}_A^{\left(j\right)}\left(a\right){\gamma }_2^{\left(i\right)}\left(m\right)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\gamma }_2^{\left(n\right)}(-mb)& =\sum _{i+j=n}({\beta }_1^{\left(i\right)}\left(b\right){\gamma }_2^{\left(j\right)}\left(m\right)+{\beta }_2^{\left(i\right)}\left(b\right){\gamma }_3^{\left(j\right)}\left(m\right)\hfill \\ & -{\gamma }_1^{\left(j\right)}\left(m\right){\beta }_2^{\left(i\right)}\left(b\right)-{\gamma }_2^{\left(j\right)}\left(m\right){\beta }_3^{\left(i\right)}\left(b\right))\hfill \\ & =\sum _{i+j=n}({\beta }_1^{\left(i\right)}\left(b\right){\gamma }_2^{\left(j\right)}\left(m\right)-{\gamma }_2^{\left(j\right)}\left(m\right){\beta }_3^{\left(i\right)}\left(b\right))\hfill \\ & =\sum _{i+j=n}{\gamma }_2^{\left(j\right)}\left(m\right)(\eta ({\beta }_1^{\left(i\right)}\left(b\right))-{\beta }_3^{\left(i\right)}\left(b\right))\hfill \\ & =-\sum _{i+j=n}{\gamma }_2^{\left(j\right)}\left(m\right)p_B^{\left(i\right)}\left(b\right)\hfill \end{array}$$

for all $a\in A,b\in B$ and $m\in M$. Hence we obtain

$$\begin{array}{cc}\hfill {\gamma }_2^{\left(n\right)}\left(am\right)& =\sum _{i+j=n}({\alpha }_1^{\left(j\right)}\left(a\right)-{\eta }^{-1}({\alpha }_3^{\left(j\right)}\left(a\right))){\gamma }_2^{\left(i\right)}\left(m\right)=\sum _{i+j=n}{p}_A^{\left(j\right)}\left(a\right){\gamma }_2^{\left(i\right)}\left(m\right)\hfill \\ \hfill {\gamma }_2^{\left(n\right)}\left(mb\right)& =-\sum _{i+j=n}{\gamma }_2^{\left(j\right)}\left(m\right)({\beta }_1^{\left(i\right)}\left(b\right)-\eta ({\beta }_3^{\left(i\right)}\left(b\right)))=\sum _{i+j=n}{\gamma }_2^{\left(j\right)}\left(m\right)p_B^{\left(i\right)}\left(b\right)\hfill \end{array}$$

(16)

for all $a\in A,b\in B,m\in M$.

On the basis of Equation (16), we propose a claim.

Claim 1: A sequence ${\mathcal{P}}_{\mathcal{A}}={\left\{p_A^{\left(n\right)}\right\}}_{n\in \mathbb{N}}$ of linear $p_A^{\left(n\right)}:A\to A$ and a sequence ${\mathcal{P}}_{\mathcal{B}}={\left\{p_B^{\left(n\right)}\right\}}_{n\in \mathbb{N}}$ of linear $p_B^{\left(n\right)}:B\to B$ are higher derivations on A and B, respectively.

By assumption (i) we have

$${\gamma }_2^{\left(n\right)}\left(a{a}^{\prime }m\right)=\sum _{i+j=n}{p}_A^{\left(i\right)}\left(a{a}^{\prime }\right){\gamma }_2^{\left(j\right)}\left(m\right)$$

for all $a,a^{\prime }\in A,m\in M$. On the other hand, we have

$$\begin{array}{cc}\hfill {\gamma }_2^{\left(n\right)}\left(a{a}^{\prime }m\right)& =\sum _{i+j=n}{p}_A^{\left(i\right)}\left(a\right){\gamma }_2^{\left(j\right)}\left(a^{\prime }m\right)\hfill \\ & =\sum _{i+j=n}{p}_A^{\left(i\right)}\left(a\right)(\sum _{j_1+j_2=j}{p}_A^{\left(j_1\right)}\left(a^{\prime }\right){\gamma }_2^{\left(j_2\right)}\left(m\right))\hfill \\ & =\sum _{i+j=n}\sum _{j_1+j_2=j}{p}_A^{\left(i\right)}\left(a\right)p_A^{\left(j_1\right)}\left(a^{\prime }\right){\gamma }_2^{\left(j_2\right)}\left(m\right)\hfill \\ & =\sum _{i+j_1+j_2=n}{p}_A^{\left(i\right)}\left(a\right)p_A^{\left(j_1\right)}\left(a^{\prime }\right){\gamma }_2^{\left(j_2\right)}\left(m\right)\hfill \\ & =\sum _{k+j_2=n}(\sum _{i+j_1=k}{p}_A^{\left(i\right)}\left(a\right)p_A^{\left(j_1\right)}\left(a^{\prime }\right)){\gamma }_2^{\left(j_2\right)}\left(m\right)\hfill \end{array}$$

for all $a,a^{\prime }\in A$ and $m\in M$.

On comparing the above two relations, we see that

$$\sum _{i+j=n}(p_A^{\left(i\right)}\left(a{a}^{\prime }\right)-\sum _{i_1+j_1=i}{p}_A^{\left(i_1\right)}\left(a\right)p_A^{\left(j_1\right)}\left(a^{\prime }\right)){\gamma }_2^{\left(j\right)}\left(m\right)=0$$

(17)

for all $a,a^{\prime }\in A,m\in M$. Next, we use mathematical induction to prove that a sequence ${\mathcal{P}}_{\mathcal{A}}={\left\{p_A^{\left(n\right)}\right\}}_{n\in \mathbb{N}}$ of the linear mapping $p_A^{\left(n\right)}:A\to A$ is a higher derivation on A. For this purpose, by the proof of [26] (Theorem 3.1) we know that for all $a,a^{\prime }\in A$, $p_A^{\left(1\right)}:A\to A$ is a derivation on A; i.e.,

$$p_A^{\left(1\right)}\left(a{a}^{\prime }\right)=p_A^{\left(1\right)}\left(a\right)a^{\prime }+a{p}_A^{\left(1\right)}\left(a^{\prime }\right)$$

whenever $n=1$. Suppose that for all $a,a^{\prime }\in A$,

$$p_A^{\left(k\right)}\left(a{a}^{\prime }\right)-\sum _{i_1+j_1=k}{p}_A^{\left(i_1\right)}\left(a\right)p_A^{\left(j_1\right)}\left(a^{\prime }\right)=0$$

whenever $1\le k\le n-1$. Taking into account (17), we therefore arrive at

$$(p_A^{\left(n\right)}\left(a{a}^{\prime }\right)-\sum _{i_1+j_1=n}{p}_A^{\left(i_1\right)}\left(a\right)p_A^{\left(j_1\right)}\left(a^{\prime }\right))m=0$$

for all $a,a^{\prime }\in A,m\in M$. Substitute $a^{″}mb$ for m in the above equation to obtain

$$(p_A^{\left(n\right)}\left(a{a}^{\prime }\right)-\sum _{i_1+j_1=n}{p}_A^{\left(i_1\right)}\left(a\right)p_A^{\left(j_1\right)}\left(a^{\prime }\right))a^{″}mb=0$$

for all $a,a^{\prime },a^{″}\in A,m\in M$ and $b\in B$. Therefore,

$$(p_A^{\left(n\right)}\left(a{a}^{\prime }\right)-\sum _{i_1+j_1=n}{p}_A^{\left(i_1\right)}\left(a\right)p_A^{\left(j_1\right)}\left(a^{\prime }\right))AMB=0$$

for all $a,a^{\prime },a^{″}\in A,m\in M$ and $b\in B$. According to the strong faithfulness of bimodule M, we have

$$p_A^{\left(n\right)}\left(a{a}^{\prime }\right)=\sum _{i_1+j_1=n}{p}_A^{\left(i_1\right)}\left(a\right)p_A^{\left(j_1\right)}\left(a^{\prime }\right)$$

for all $a,a^{\prime }\in A$ and so a sequence ${\mathcal{P}}_{\mathcal{A}}={\left\{p_A^{\left(n\right)}\right\}}_{n\in \mathbb{N}}$ of the linear mapping $p_A^{\left(n\right)}:A\to A$ is a higher derivation on A. Adopting the same discussion as for relations $P_A^{\left(n\right)}$ and the proof of [26] (Theorem 4.4), we can prove that a sequence ${\mathcal{P}}_{\mathcal{B}}={\left\{p_B^{\left(n\right)}\right\}}_{n\in \mathbb{N}}$ of linear mappings $P_B^{\left(n\right)}:B\to B$ is a higher derivation on B.

Let us check

$$\left[\begin{array}{cc}0& {\alpha }_2^{\left(n\right)}\left(a{a}^{\prime }\right)-\sum _{i+j=n}{p}_A^{\left(i\right)}\left(a\right){\alpha }_2^{\left(j\right)}\left(a^{\prime }\right)\\ 0& 0\end{array}\right]\in {\mathcal{C}}_2\left(\mathcal{T}\right)$$

(18)

and

$$\left[\begin{array}{cc}0& {\beta }_2^{\left(n\right)}\left(b{b}^{\prime }\right)-\sum _{i+j=n}{\beta }_2^{\left(i\right)}\left(b\right)p_B^{\left(j\right)}\left(b^{\prime }\right)\\ 0& 0\end{array}\right]\in {\mathcal{C}}_2\left(\mathcal{T}\right)$$

(19)

for all $a,a^{\prime }\in A$ and $b,b^{\prime }\in B$ by complete induction on n.

Let us prove the relationship (18) by mathematical induction for n. When $n=1$, due to the proof of [26] (Theorem 3.2), it is clear that

$${\alpha }_2^{\left(1\right)}\left(a{a}^{\prime }\right)-a{\alpha }_2^{\left(1\right)}\left(a^{\prime }\right)={\alpha }_2^{\left(1\right)}\left(a{a}^{\prime }\right)-a{\alpha }_2^{\left(1\right)}\left(a^{\prime }\right)-p_A^{\left(1\right)}\left(a\right){\alpha }_2^{\left(0\right)}\left(a^{\prime }\right)\in {\mathcal{C}}_2\left(T\right)$$

with the help of the definition of Lie higher derivation for all $a,a^{\prime }\in A$.

Suppose that

$${\alpha }_2^{\left(k\right)}\left(a{a}^{\prime }\right)-\sum _{i+j=k}{p}_A^{\left(i\right)}\left(a\right){\alpha }_2^{\left(j\right)}\left(a^{\prime }\right)\in {\mathcal{C}}_2\left(\mathcal{T}\right)$$

for all $1\le k\le n-1$; i.e.,

$$({\alpha }_2^{\left(k\right)}\left(a{a}^{\prime }\right)-\sum _{i+j=k}{p}_A^{\left(i\right)}\left(a\right){\alpha }_2^{\left(j\right)}\left(a^{\prime }\right))B=0=A({\alpha }_2^{\left(k\right)}\left(a{a}^{\prime }\right)-\sum _{i+j=k}{p}_A^{\left(i\right)}\left(a\right){\alpha }_2^{\left(j\right)}\left(a^{\prime }\right)).$$

With the help of (6) and replacing a by $a{a}^{\prime }$, we have

$$\begin{array}{cc}& \sum _{i+j=n}{\alpha }_2^{\left(i\right)}\left(a{a}^{\prime }\right)p_B^{\left(j\right)}\left(b\right)=\sum _{i+j=n}{p}_A^{\left(i\right)}\left(a{a}^{\prime }\right){\beta }_2^{\left(j\right)}\left(b\right)\hfill \\ & =\sum _{i+j=n}(\sum _{i_1+i_2=i}{p}_A^{\left(i_1\right)}\left(a\right)p_A^{\left(i_2\right)}\left(a^{\prime }\right)){\beta }_2^{\left(j\right)}\left(b\right)\hfill \\ & =\sum _{i+j=n}\sum _{i_1+i_2=i}{p}_A^{\left(i_1\right)}\left(a\right)p_A^{\left(i_2\right)}\left(a^{\prime }\right){\beta }_2^{\left(j\right)}\left(b\right)\hfill \\ & =\sum _{i_1+i_2+j=n}{p}_A^{\left(i_1\right)}\left(a\right)p_A^{\left(i_2\right)}\left(a^{\prime }\right){\beta }_2^{\left(j\right)}\left(b\right)\hfill \\ & =\sum _{i_1+s=n}{p}_A^{\left(i_1\right)}\left(a\right)(\sum _{i_2+j=s}{p}_A^{\left(i_2\right)}\left(a^{\prime }\right){\beta }_2\left(j\right)\left(b\right))\hfill \\ & =\sum _{i_1+s=n}{p}_A^{\left(i_1\right)}\left(a\right)(\sum _{i_2+j=s}{\alpha }_2^{\left(i_2\right)}\left(a^{\prime }\right)p_B^{\left(j\right)}\left(b\right))\hfill \\ & =\sum _{i_1+i_2+j=n}{p}_A^{\left(i_1\right)}\left(a\right){\alpha }_2^{\left(i_2\right)}\left(a^{\prime }\right))p_B^{\left(j\right)}\left(b\right)\hfill \\ & =\sum _{i+j=n}(\sum _{i_1+i_2=i}{p}_A^{\left(i_1\right)}\left(a\right){\alpha }_2^{\left(i_2\right)}\left(a^{\prime }\right))p_B^{\left(j\right)}\left(b\right)\hfill \end{array}$$

for all $a,a^{\prime }\in A$ and $b\in B$. Furthermore, we have

$$\begin{array}{cc}\hfill 0=& \sum _{i+j=n}({\alpha }_2^{\left(i\right)}\left(a{a}^{\prime }\right)-\sum _{i_1+i_2=i}{p}_A^{\left(i_1\right)}\left(a\right){\alpha }_2^{\left(i_2\right)}\left(a^{\prime }\right))p_B^{\left(j\right)}\left(b\right)\hfill \\ \hfill =& \sum _{i+j=n,i\ne n}({\alpha }_1^{\left(i\right)}\left(a{a}^{\prime }\right)-\sum _{i_1+i_2=i}{p}_A^{\left(i_1\right)}\left(a\right){\alpha }_2^{\left(i_2\right)}\left(a^{\prime }\right))p_B^{\left(j\right)}\left(b\right)\hfill \\ & +({\alpha }_2^{\left(n\right)}\left(a{a}^{\prime }\right)-\sum _{i_1+i_2=n}{p}_A^{\left(i_1\right)}\left(a\right){\alpha }_2^{\left(i_2\right)}\left(a^{\prime }\right))b\hfill \end{array}$$

for all $a,a^{\prime }\in A$ and $b\in B$. Taking into account the above equation and inductive hypothesis, we have

$$({\alpha }_2^{\left(n\right)}\left(a{a}^{\prime }\right)-\sum _{i+j=n}{p}_A^{\left(i\right)}\left(a\right){\alpha }_2^{\left(j\right)}\left(a^{\prime }\right))B=0$$

for all $a,a^{\prime }\in A$. The condition (iii) implies that

$$A({\alpha }_2^{\left(n\right)}\left(a{a}^{\prime }\right)-\sum _{i+j=n}{p}_A^{\left(i\right)}\left(a\right){\alpha }_2^{\left(j\right)}\left(a^{\prime }\right))=0$$

for all $a,a^{\prime }\in A$. In view of the extreme center ${\mathcal{C}}_2\left(\mathcal{A}\right)$, we see that

$$\left[\begin{array}{cc}0& {\alpha }_2^{\left(n\right)}\left(a{a}^{\prime }\right)-\sum _{i+j=n}{p}_A^{\left(i\right)}\left(a\right){\alpha }_2^{\left(j\right)}\left(a^{\prime }\right)\\ 0& 0\end{array}\right]\in {\mathcal{C}}_2\left(\mathcal{T}\right)$$

for all $a,a^{\prime }\in A$. Similarly, with the help of [26] (Theorem 3.2) and (6), we can conclude that (19) is true.

We define a mapping ${\tau }_n:\mathcal{T}\to \mathcal{T}$ in the following way:

$${\tau }_n\left(\left[\begin{array}{cc}a& m\\ 0& b\end{array}\right]\right)=\left[\begin{array}{cc}{\phi }^{-1}\left({\alpha }_3^{\left(n\right)}\left(a\right)\right)+{\beta }_2^{\left(n\right)}\left(b\right)+{\gamma }_1^{\left(n\right)}\left(m\right)& 0\\ 0& \phi \left({\beta }_2^{\left(n\right)}\left(b\right)\right)+{\alpha }_3^{\left(n\right)}\left(a\right)+{\gamma }_3^{\left(n\right)}\left(m\right)\end{array}\right]$$

for all $a\in A,b\in B,m\in M$. We note that ${\tau }_n:\mathcal{T}\to {\mathcal{C}}_1\left(\mathcal{T}\right)$ for arbitrary $n\in \mathbb{N}$.

In order to obtain the conclusion of this theorem, we divide its proof into two different cases.

Case 1: Suppose first that

$${\alpha }_2^{\left(n\right)}\left(a{a}^{\prime }\right)-\sum _{i+j=n}{p}_A^{\left(i\right)}\left(a\right){\alpha }_2^{\left(j\right)}\left(a^{\prime }\right)-{\beta }_2^{\left(n\right)}\left(b{b}^{\prime }\right)+\sum _{i+j=n}{\beta }_2^{\left(i\right)}\left(b\right)p_B^{\left(j\right)}\left(b^{\prime }\right)=0$$

for all $a,a^{\prime }\in A,b,b^{\prime }\in B$. This implies that

$${\alpha }_2^{\left(n\right)}\left(a{a}^{\prime }\right)-\sum _{i+j=n}{p}_A^{\left(i\right)}\left(a\right){\alpha }_2^{\left(j\right)}\left(a^{\prime }\right)=0\mathrm{and}{\beta }_2^{\left(n\right)}\left(b{b}^{\prime }\right)-\sum _{i+j=n}{\beta }_2^{\left(i\right)}\left(b\right)p_B^{\left(j\right)}\left(b^{\prime }\right)=0$$

(20)

for all $a,a^{\prime }\in A$ and $b,b^{\prime }\in B$.

Two mappings are defined as follows:

(1)

Define the first mapping ${\delta }_n:\mathcal{T}\to \mathcal{T}$ as

$$\begin{array}{cc}\hfill {\delta }_n\left(\left[\begin{array}{cc}a& m\\ 0& b\end{array}\right]\right)& =\left[\begin{array}{cc}{\alpha }_1^{\left(n\right)}\left(a\right)-{\eta }^{-1}\left({\alpha }_3^{\left(n\right)}\left(a\right)\right)& {\alpha }_2^{\left(n\right)}\left(a\right)+{\beta }_2^{\left(n\right)}\left(b\right)+{\gamma }_2^{\left(n\right)}\left(m\right)\\ 0& {\beta }_3^{\left(n\right)}\left(b\right)-\eta \left({\beta }_1^{\left(n\right)}\left(b\right)\right)\end{array}\right]\hfill \\ & =\left[\begin{array}{cc}{p}_A^{\left(n\right)}\left(a\right)& {\alpha }_2^{\left(n\right)}\left(a\right)+{\beta }_2^{\left(n\right)}\left(b\right)+{\gamma }_2^{\left(n\right)}\left(m\right)\\ 0& p_B^{\left(n\right)}\left(b\right)\end{array}\right]\hfill \end{array}$$

for all $a\in A,b\in B,m\in M$. Using Claim 1 and together with (16), (18), and (20), we get from Theorem 2 that a sequence $\mathrm{\Delta }={\left\{{\delta }_n\right\}}_{n\in \mathbb{N}}$ of linear mappings ${\delta }_n:\mathcal{T}\to \mathcal{T}$ is a higher derivation.

(2)

Let us define the second mapping ${\sigma }_n:\mathcal{T}\to \mathcal{T}$ as follows:

$${\sigma }_n\left(\left[\begin{array}{cc}a& m\\ 0& b\end{array}\right]\right)=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$$

for all $a\in A,b\in B$ and $m\in M$.

It is clear that $L_n={\delta }_n+{\sigma }_n+{\tau }_n$. Using (16), (18), and (19), we obtain from Lemma 1 that $\mathrm{\Delta }={\left\{{\delta }_n\right\}}_{n\in \mathbb{N}}$ is a higher derivation on $\mathcal{T}$. We easily check that ${\tau }_n\left([x,y]\right)=0$ for all $x,y\in \mathcal{T}$.

Case 2: Suppose next that

$${\alpha }_2^{\left(n\right)}\left(a_0{a}_0^{\prime }\right)-\sum _{i+j=n}{p}_A^{\left(i\right)}\left(a_0\right){\alpha }_2^{\left(j\right)}\left(a_0^{\prime }\right)-{\beta }_2^{\left(n\right)}\left(b_0{b}_0^{\prime }\right)+\sum _{i+j=n}{\beta }_2^{\left(i\right)}\left(b_0\right)p_B^{\left(j\right)}\left(b_0^{\prime }\right)\ne 0$$

(21)

for some $a_0,a_0^{\prime }\in A$ and $b_0,b_0^{\prime }\in B$. Let us define the following two mappings:

(1)

Define the first mapping ${\delta }_n:\mathcal{T}\to \mathcal{T}$ as

$${\delta }_n\left(\left[\begin{array}{cc}a& m\\ 0& b\end{array}\right]\right)=\left[\begin{array}{cc}{p}_A^{\left(n\right)}\left(a\right)& {\gamma }_3^{\left(n\right)}\left(m\right)\\ 0& p_B^{\left(n\right)}\left(b\right)\end{array}\right]$$

for all $a\in A,b\in B,m\in M$. It follows from (6) and Claim 1 that a sequence $\mathrm{\Delta }={\left\{{\delta }_n\right\}}_{n\in \mathbb{N}}$ of linear mappings ${\delta }_n:\mathcal{T}\to \mathcal{T}$ is a higher derivation.

(2)

Let us define the second mapping ${\sigma }_n:\mathcal{T}\to \mathcal{T}$ as follows:

$${\sigma }_n\left(\left[\begin{array}{cc}a& m\\ 0& b\end{array}\right]\right)=\left[\begin{array}{cc}0& {\alpha }_2^{\left(n\right)}\left(a\right)+{\beta }_2^{\left(n\right)}\left(b\right)\\ 0& 0\end{array}\right]$$

for all $a\in A,b\in B,m\in M$.

It is clear that $L_n={\delta }_n+{\sigma }_n+{\tau }_n$. Using (22) we obtain from Theorem 1 that ${\delta }_n$ is a higher derivation on $\mathcal{T}$.

Below we prove in the form of Claim 2 that ${\left\{{\sigma }_n\right\}}_{n\in N}$ of the linear mapping ${\sigma }_n:\mathcal{T}\to \mathcal{T}$ is an extreme Lie higher derivation at zero products and that the ${\left\{{\sigma }_n\right\}}_{n\in N}$ of the linear mapping ${\sigma }_n:\mathcal{T}\to \mathcal{T}$ satisfies the relation ${\tau }_n\left([x,y]\right)=0$ with $xy=0$. We propose the second claim.

Claim 2: With the same notations as above, we have that a sequence $\sum ={\left\{{\sigma }_n\right\}}_{n\in \mathcal{N}}$ of the mappings ${\sigma }_n:\mathcal{T}\to \mathcal{T}$ and ${\tau }_n$ satisfy the following relations:

(a)

${\sigma }_n$ is a nonzero mapping on $\mathcal{T}$;

(b)

${\sigma }_n\left(xy\right)-{\sum }_{i+j=n}({\delta }_i\left(x\right){\sigma }_j\left(y\right)+{\sigma }_i\left(x\right){\delta }_j\left(y\right))\in {\mathcal{C}}_2\left(\mathcal{T}\right)$ for all $x,y\in \mathcal{T}$;

(c)

${\sigma }_n\left([x,y]\right)={\sum }_{i+j=n}([{\delta }_i\left(x\right),{\sigma }_j\left(y\right)]+[{\sigma }_i\left(x\right),{\delta }_j\left(y\right)])$ for all $x,y\in \mathcal{T}$;

(d)

${\tau }_n\left([x,y]\right)=0$ for all $x,y\in \mathcal{T}$ with $xy=0$.

Indeed, we first prove conclusion $\left(b\right)$.

According to (18) and (19), there exists $m_0,m_0^{\prime }\in M$ satisfying the following relation:

$$\begin{array}{cc}\hfill & {\alpha }_2^{\left(n\right)}\left(a{a}^{\prime }\right)=\sum _{i+j=n}{p}_A^{\left(i\right)}\left(a\right){\alpha }_1^{\left(j\right)}\left(a^{\prime }\right)+m_0\hfill \\ \hfill & \mathrm{and}\hfill \\ \hfill & {\beta }_2^{\left(n\right)}\left(b{b}^{\prime }\right)=\sum _{i+j=n}{\beta }_2^{\left(i\right)}\left(b\right)p_B^{\left(j\right)}\left(b^{\prime }\right)+m_0^{\prime }\hfill \end{array}$$

(22)

for all $a,a^{\prime }\in A$ and $b,b^{\prime }\in B$ and some $m_0,m_0^{\prime }\in M$.

For arbitrary $x=a+m+b$ and $y=a^{\prime }+m^{\prime }+b^{\prime }$, using (6) and (22) we have

$$\begin{array}{cc}\hfill {\sigma }_n\left(xy\right)& ={\sigma }_n\left(\left[\begin{array}{cc}a& m\\ 0& b\end{array}\right]\left[\begin{array}{cc}{a}^{\prime }& m^{\prime }\\ 0& b^{\prime }\end{array}\right]\right)\hfill \\ & ={\sigma }_n\left(\left[\begin{array}{cc}a{a}^{\prime }& a{m}^{\prime }+m{b}^{\prime }\\ 0& b{b}^{\prime }\end{array}\right]\right)=\left[\begin{array}{cc}0& {\alpha }_2^{\left(n\right)}\left(a{a}^{\prime }\right)+{\beta }_2^{\left(n\right)}\left(b{b}^{\prime }\right)\\ 0& 0\end{array}\right]\hfill \\ & =\sum _{i+j=n}\left[\begin{array}{cc}0& p_A^{\left(i\right)}\left(a\right){\alpha }_2^{\left(j\right)}\left(a^{\prime }\right)+{\beta }_2^{\left(i\right)}\left(b\right)p_B^{\left(j\right)}\left(b^{\prime }\right)\\ 0& 0\end{array}\right]+\left[\begin{array}{cc}0& m_0+m_0^{\prime }\\ 0& 0\end{array}\right]\hfill \\ & =\sum _{i+j=n}\left[\begin{array}{cc}0& p_A^{\left(i\right)}\left(a\right){\alpha }_2^{\left(j\right)}\left(a^{\prime }\right)+p_A^{\left(i\right)}\left(a\right){\beta }_2^{\left(j\right)}\left(b\right)+{\alpha }_2^{\left(i\right)}\left(a\right)p_B^{\left(j\right)}\left(b\right)+{\beta }_2^{\left(i\right)}\left(b\right)p_B^{\left(j\right)}\left(b^{\prime }\right)\\ 0& 0\end{array}\right]+\left[\begin{array}{cc}0& m_0+m_0^{\prime }\\ 0& 0\end{array}\right]\hfill \\ & =\sum _{i+j=n}({\delta }_i\left(x\right){\sigma }_j\left(y\right)+{\sigma }_i\left(x\right){\delta }_j\left(y\right))+\left[\begin{array}{cc}0& m_0+m_0^{\prime }\\ 0& 0\end{array}\right].\hfill \end{array}$$

Therefore, we have

$${\sigma }_n\left(xy\right)-\sum _{i+j=n}({\delta }_i\left(x\right){\sigma }_j\left(y\right)+{\sigma }_i\left(x\right){\delta }_j\left(y\right))\in {\mathcal{C}}_2\left(\mathcal{T}\right)$$

for all $x,y\in \mathcal{T}$.

In view of (21) and the last relation, we see that

$$\begin{array}{cc}\hfill {\sigma }_n\left(\left[\begin{array}{cc}{a}_0& m_0\\ 0& b_0\end{array}\right]\left[\begin{array}{cc}{a}_0^{\prime }& m_0^{\prime }\\ 0& b_0^{\prime }\end{array}\right]\right)& -\sum _{i+j=n}({\delta }_i\left(\left[\begin{array}{cc}{a}_0& m_0\\ 0& b_0\end{array}\right]\right){\sigma }_j\left(\left[\begin{array}{cc}{a}_0^{\prime }& m_0^{\prime }\\ 0& b_0^{\prime }\end{array}\right]\right)\hfill \\ & -{\sigma }_i\left(\left[\begin{array}{cc}{a}_0& m_0\\ 0& b_0\end{array}\right]\right){\delta }_j\left(\left[\begin{array}{cc}{a}_0^{\prime }& m_0^{\prime }\\ 0& b_0^{\prime }\end{array}\right]\right))\ne 0.\hfill \end{array}$$

We see that ${\sigma }_n$ is not a nonzero mapping.

In order to prove conclusions $\left(c\right)$ and $\left(d\right)$, set

$${\mu }_n(x,y)={\sigma }_n\left([x,y]\right)-\sum _{i+j=n}([{\delta }_i\left(x\right),{\sigma }_j\left(y\right)]+[{\sigma }_i\left(x\right),{\delta }_j\left(y\right)])$$

for all $x,y\in \mathcal{T}$ with $xy=0$. Note ${\mu }_n(x,y)\in {\mathcal{C}}_2\left(\mathcal{T}\right)$. Since $L_n={\delta }_n+{\sigma }_n+{\tau }_n$ is a Lie higher derivation at zero products, we have

$$\begin{array}{cc}\hfill L_n\left([x,y]\right)& =\sum _{i+j=n}[L_i\left(x\right),L_i\left(y\right)]\hfill \\ & =\sum _{i+j=n}[{\delta }_i\left(x\right)+{\sigma }_i\left(x\right)+{\tau }_i\left(x\right),{\delta }_j\left(y\right)+{\sigma }_j\left(y\right)+{\tau }_j\left(y\right)]\hfill \\ & =\sum _{i+j=n}[{\delta }_i\left(x\right)+{\sigma }_i\left(x\right),{\delta }_j\left(y\right)+{\sigma }_j\left(y\right)]\hfill \\ & =\sum _{i+j=n}([{\delta }_i\left(x\right),{\delta }_j\left(y\right)]+[{\delta }_i\left(x\right),{\sigma }_j\left(y\right)]+[{\sigma }_i\left(x\right),{\delta }_j\left(y\right)])\hfill \\ & =L_n\left([x,y]\right)-{\sigma }_n\left([x,y]\right)+{\mu }_n(x,y)\hfill \end{array}$$

for all $x,y\in \mathcal{T}$ with $xy=0$. This implies that

$${\sigma }_n\left([x,y]\right)+{\mu }_n(x,y)=0$$

for all $x,y\in \mathcal{T}$ with $xy=0$. Since ${\mathcal{C}}_1\left(\mathcal{T}\right)\bigcap {\mathcal{C}}_2\left(\mathcal{T}\right)=0$, we get that

$${\tau }_n\left([x,y]\right)=0={\mu }_n(x,y)$$

for all $x,y\in \mathcal{T}$ with $xy=0$. So far, we have proven that ${\left\{{\sigma }_n\right\}}_{n\in N}$ of the linear mapping ${\sigma }_n:\mathcal{T}\to \mathcal{T}$ is an extreme non-global Lie higher derivation and that the ${\left\{{\sigma }_n\right\}}_{n\in N}$ of the linear mapping ${\sigma }_n:\mathcal{T}\to \mathcal{T}$ satisfies the relation ${\tau }_n\left([x,y]\right)=0$ with $xy=0$. That is, Claim 2 is true. □

Remark 1.

When $n=1$, a sequence $\sum ={\left\{{\sigma }_n\right\}}_{n\in \mathbb{N}}$ of the linear mapping ${\sigma }_n:\mathcal{T}\to \mathcal{T}$ will degenerate into an extreme Lie derivation at zero products introduced by Behfar and Ghahramani [26] (Definition 4.2).

As a consequence we have the following results by Wang (see [26] (Theorem 4.3)).

Corollary 1

([26] (Theorem 4.3)). Let $\mathcal{T}=\left[\begin{array}{cc}A& M\\ O& B\end{array}\right]$ be a triangular algebra, if the following statements hold true:

(i) 

$\mathcal{C}\left(A\right)={\pi }_A\left(\mathcal{T}\right)$ and $\mathcal{C}\left(B\right)={\pi }_B\left(\mathcal{T}\right);$

(ii) 

M is strong faithful;

(iii) 

For any $m_0\in M$, we have that $A{m}_0=0$ if and only if $m_{0}B=0$.

Let a linear mapping $L:\mathcal{T}\to \mathcal{T}$ be a non-global Lie derivation. Then there exists a derivation δ, an extreme non-global Lie derivation σ, and an $\mathcal{R}$-linear mapping $\tau :\mathcal{T}\to \mathcal{C}\left(\mathcal{T}\right)$ vanishing on commutators $[x,y]$ with $xy=0$ such that

$$L=\delta +\sigma +\tau$$

for all $x,y\in \mathcal{T}$.

In this paper, we only assume that the triangular algebra does not contain the identity element; that is, in the whole proof process, this paper is characterized by not using the identity element, but for the triangular algebra with the identity element, we can obtain the following corollaries.

Corollary 2

([22] (Theorem 2.2)). Let $\mathcal{T}=\left[\begin{array}{cc}A& M\\ O& B\end{array}\right]$ be a triangular algebra, if the following statements hold true:

(i) 

$\mathcal{C}\left(\mathcal{A}\right)={\pi }_{\mathcal{A}}\left(\mathcal{C}\left(\mathcal{T}\right)\right)$ and $\mathcal{C}\left(\mathcal{B}\right)={\pi }_{\mathcal{B}}\left(\mathcal{C}\left(\mathcal{T}\right)\right)$;

(ii) 

M is faithful;

(iii) 

Both A and B are unital.

Let a sequence $\mathbb{L}={\left\{L_n\right\}}_{n\in \mathbb{N}}$ of linear mappings ${\delta }_n:\mathcal{T}\to \mathcal{T}$ be a non-global Lie higher derivation if and only if there exists a higher derivation $\mathrm{\Delta }={\left\{{\delta }_n\right\}}_{n\in \mathbb{N}}$ and an $\mathcal{R}$-linear mapping ${\tau }_n:\mathcal{T}\to \mathcal{C}\left(\mathcal{T}\right)$ vanishing on commutators $[x,y]$ with $xy=0$ such that

$$L_n\left(x\right)={\delta }_n\left(x\right)+{\tau }_n\left(x\right)$$

for all $x\in \mathcal{T}$.

As non-global Lie higher derivations generalize non-global Lie derivations, Corollary 3 is a straightforward consequence of Corollary 2.

Corollary 3

([21] (Theorem 2.1)). Let $\mathcal{T}=\left[\begin{array}{cc}A& M\\ O& B\end{array}\right]$ be a triangular algebra, if the following statements hold true:

(i) 

$\mathcal{C}\left(\mathcal{A}\right)={\pi }_{\mathcal{A}}\left(\mathcal{C}\left(\mathcal{T}\right)\right)$ and $\mathcal{C}\left(\mathcal{B}\right)={\pi }_{\mathcal{B}}\left(\mathcal{C}\left(\mathcal{T}\right)\right)$;

(ii) 

M is faithful;

(iii) 

Both A and B are unital.

Let $\delta :\mathcal{T}\to \mathcal{T}$ be an $\mathcal{R}$-linear mapping such that

$$\delta \left(\right[x,y\left]\right)=\left[\delta \right(x),y]+[x,\delta (y\left)\right]$$

for any $x,y\in \mathcal{T}$ with $xy=0$; then there exists a derivation d of $\mathcal{T}$ and an $\mathcal{R}$-linear mapping $\tau :\mathcal{T}\to \mathcal{C}\left(\mathcal{T}\right)$ vanishing at commutators $[x,y]$ with $xy=0$ such that

$$\delta \left(x\right)=d\left(x\right)+\tau \left(x\right),\mathit{for}\mathit{all} x\in \mathcal{T}.$$

In addition to the above corollaries, Theorem 2 has important applications in typical examples, such as upper triangular matrix algebras $T_n\left(\mathcal{A}\right)(n\ge 2)$ defined on faithful algebras $\mathcal{A}$ (see [4] (Theorem 4.1) for definition). With the help of [4] (Theorem 4.1), $T_n\left(\mathcal{A}\right)$ can be viewed as the triangular algebra

$$\left[\begin{array}{cc}\mathcal{A}& {\mathcal{A}}^{n-1}\\ 0& T_{n-1}\left(\mathcal{A}\right)\end{array}\right]:=\left[\begin{array}{cc}A& M\\ 0& B\end{array}\right].$$

Through the medium of [4] (Theorem 4.1), the upper triangular matrix algebra $T_n\left(\mathcal{A}\right)(n\ge 2)$ coincides with the conditions of Theorem 2.

According to [4] (4. An application), the algebra $T_n\left(\mathcal{A}\right)$ satisfies the requirements of Theorem 2 whenever $\mathcal{A}$ is faithful. This fact, together with the results of [4] (Theorem 4.1), ensures the applicability of Corollary 4, which we thus deduce and present below.

Corollary 4.

Let $\mathcal{A}$ be a faithful algebra and $T_n\left(\mathcal{A}\right)$ the upper triangular matrix algebra of degree $n\ge 2$. Then every non-global Lie higher derivation $\mathbb{L}={L_n}_{n\in \mathbb{N}}$ on $T_n\left(\mathcal{A}\right)$ admits a decomposition

$$L_n={\delta }_n+{\tau }_n$$

where ${{\delta }_n}_{n\in \mathbb{N}}$ is a higher derivation and each ${\tau }_n:T_n\left(\mathcal{A}\right)\to \mathcal{C}\left(T_n\left(\mathcal{A}\right)\right)$ is an $\mathcal{R}$-linear map vanishing on commutators $[x,y]$ whenever $xy=0$.

As established in [4] (4. An application), all semiprime algebras are faithful. Together with the results of [4] (Corollary 4.1), this directly implies the validity of Corollary 5:

Corollary 5.

Let $n\ge 2$ and consider the upper triangular matrix algebra $T_n\left(\mathcal{A}\right)$ over a semiprime algebra $\mathcal{A}$. Suppose $\mathbb{L}={L_n}_{n\in \mathbb{N}}$ is a sequence of linear mappings on $T_n\left(\mathcal{A}\right)$ forming a non-global Lie higher derivation. Then $\mathbb{L}$ can be expressed as

$$L_n={\delta }_n+{\tau }_n$$

where $\mathrm{\Delta }={\delta }_{n}n\in \mathbb{N}$ is a higher derivation and ${\tau }_{n}n\in \mathbb{N}$ is a sequence of $\mathcal{R}$-linear mappings into the center $\mathcal{C}\left(T_n\left(\mathcal{A}\right)\right)$ that vanish on all commutators $[x,y]$ satisfying $xy=0$.

The applicability of Corollary 5 is demonstrated by considering the wide range of semiprime algebras presented in [27] (Example 2.5), encompassing double affine Hecke algebras, graded Hecke algebras, rational Cherednik algebras, Iwasawa algebras, algebras of bounded linear operators, and von Neumann algebras. This application directly leads to numerous conclusions structurally similar to Corollary 5.

4. Open Problems and Future Work

The research topic of this paper is to characterize the structure of non-global Lie higher derivations on triangular algebras without assuming the existence of idempotents. Under certain conditions, every non-global Lie higher derivation can be decomposed as the sum of a higher derivation, an extreme non-global Lie higher derivation, and a central mapping. This structure provides a unified characterization of non-global Lie higher derivations on triangular algebras, both with and without idempotents. However, it is important to note that a key prerequisite for this conclusion is that the triangular algebra must be associative, and this condition may vary depending on whether the algebra has a unit element.

In recent years, the study of the structure of various linear or nonlinear mappings on non-associative rings or algebras has attracted considerable scholarly attention. Among these, alternative rings (or algebras), as investigated by Ferreira and collaborators, have been a prominent class of non-associative algebras $\mathfrak{A}$. For details on their definition, refer to [30]. The structure of linear or nonlinear mappings on alternative rings (or algebras) has been extensively studied, including Lie multiplicative derivations [31], nonlinear mixed ∗-Jordan-type derivations [32], Jordan Elementary Maps [33], ∗-Lie-type maps [34], commuting maps [35], Lie triple centralizers [30], and so on. In 2019, Ferreira and collaborators [30] studied the structure of Lie multiplicative derivations on alternative rings $\mathfrak{A}$, proving that every Lie multiplicative derivation is the sum of an additive derivation and a central mapping. Subsequently, in 2025, Ferreira and collaborators [36] investigated the problem of additive maps preserving products equal to fixed elements. Inspired by [26,30,36], we hereby propose an open question:

Question 2.

What is the structure of non-global Lie multiplicative derivations on an alternative ring $\mathfrak{A}$? Specifically, when can such a derivation be decomposed into the sum of an additive derivation and a central map that annihilates every commutator $[x,y]$ with $xy=0$?

5. Conclusions

The central result of this paper characterizes the structure of non-global Lie higher derivations on triangular algebras without assuming an identity element. Under specified conditions, we prove that each such derivation admits a canonical decomposition into the sum of three components: a higher derivation, an extreme non-global Lie higher derivation, and a quasi-derivation subject to specific constraints. The explicit structure is detailed in the following.

Theorem 3.

Let $\mathcal{T}=\left[\begin{array}{cc}A& M\\ O& B\end{array}\right]$ be a triangular algebra, if the following statements hold true:

(i) 

$\mathcal{C}\left(\mathcal{A}\right)={\pi }_{\mathcal{A}}\left(\mathcal{C}\left(\mathcal{T}\right)\right)$ and $\mathcal{C}\left(\mathcal{B}\right)={\pi }_{\mathcal{B}}\left(\mathcal{C}\left(\mathcal{T}\right)\right)$;

(ii) 

M is strong faithful;

(iii) 

For any $m_0\in M$, we have that $A{m}_0=0$ if and only if $m_{0}B=0$.

Let a sequence $\mathbb{L}={\left\{L_n\right\}}_{n\in \mathbb{N}}$ of linear mappings $L_n:\mathcal{T}\to \mathcal{T}$ be a non-global Lie higher derivation. Then there exists a higher derivation $\mathrm{\Delta }={\left\{{\delta }_n\right\}}_{n\in \mathbb{N}}$ of the linear mapping ${\delta }_n:\mathcal{T}\to \mathcal{T}$, an extreme non-global Lie higher derivation $\sum ={\left\{{\sigma }_n\right\}}_{n\in \mathbb{N}}$ of the linear mapping ${\sigma }_n:\mathcal{T}\to \mathcal{T}$, and an $\mathcal{R}$-linear mapping ${\tau }_n:\mathcal{T}\to {\mathcal{C}}_1\left(\mathcal{T}\right)$ vanishing on commutators $[x,y]$ with $xy=0$ such that

$$L_n\left(x\right)={\delta }_n\left(x\right)+{\sigma }_n\left(x\right)+{\tau }_n\left(x\right)$$

for all $x\in \mathcal{T}$.