Additivity of nonlinear higher anti-derivable mappings on generalized matrix algebras

1.   Introduction

In past decades, research of derivations and nonlinear derivable mappings on algebras has attracted the attention of many mathematicians.

Definition 1.1 Let $\mathcal{R}$ be a commutative ring with identity and $\mathcal{A}$ a unital algebra over $\mathcal{R}$, and ${\mathbb{N}}$ the set of non-negative integers, $i, j, k, n\in {\mathbb{N}}$.

(1) If $\Delta$ is an additive mapping such that

for all $X, Y\in\mathcal{A}$, then $\Delta$ is said to be a derivation. If $\Delta$ is not necessarily additive and Eq (1.1) hold for all $X, Y\in\mathcal{A}$, then $\Delta$ is said to be a nonlinear derivable mapping.

(2) For any $X, Y\in \mathcal{A}$, call $X\circ Y: = XY+YX$ the Jordan product of $X, Y$. If $\Delta$ is an additive mapping such that

for all $X, Y\in\mathcal{A}$, then $\Delta$ is said to be a Jordan derivation. Nonlinear Jordan derivation is defined similarly to the nonlinear derivable mapping.

(3) Let $D = \{d_{n}\}_{n\in \mathbb{N}}$ be a sequence of additive mappings (resp., without assumption of additivity) on $\mathcal{A}$ with $d_{0} = id_{\mathcal{A}}$ the identity mapping on $\mathcal{A}$ such that

for all $n\in\mathbb{N}$, and $X, Y\in\mathcal{A}$, then $D$ is said to be a higher derivation (resp., nonlinear higher derivable mapping).

Obviously, every additive derivation is an additive Jordan derivation, and every additive higher derivation is an additive Jordan higher derivation. However, the inverse statement is not true in general. It is natural to ask the following two questions:

Problem 1 Under what conditions is a Jordan (higher) derivation is a (higher) derivation?

Problem 2 Under what conditions is a nonlinear (Jordan, higher) derivation is a (Jordan, higher) derivation?

There are many works that consider Problem 1. For example, see ^[1,2,3,4,5]. In this paper, we focus on Problem 2. Rickart ^[6] proved that, under certain conditions, any one-to-one and multiplicative mapping from a ring into another ring is necessarily additive. Martindale ^[7] obtained the result that each multiplicative bijective mapping on an arbitrary algebra which contains a nontrivial idempotent is automatically additive. For other similar results about additivity of multiplicative mappings on rings or algebras, we refer the readers to ^[8,9,10,11] and references therein for more details. Daif ^[12] showed that, under certain conditions, any multiplicative derivation is additive. Later, Daif ^[13] extended this result to the case of multiplicative generalized derivation. Lu ^[14] proved that, under some conditions, every multiplicative Jordan derivation on a prime ring is an additive derivation. For more similar results about additivity of nonlinear Jordan derivable on rings or algebras, see ^[15,16] and references therein. Fu and Xiao ^[17] and Ashraf and Jabeen ^[18] showed that all nonlinear Jordan higher derivable mappings and nonlinear Jordan higher triple derivable mappings on triangular algebras is an additive higher derivation, respectively.

In ^[1], Benkovič defined anti-derivations on algebras as the following.

Definition 1.2 Let $C$ be a commutative ring with unity, $A$ an algebra over $C$ and $M$ an $A$-bimodule. Let $\delta:A\to M$ be a linear map. If

for all $a, b\in A$, then $\delta$ is said an anti-derivation. For more results about anti-derivation on rings or algebras, see ^[19,20] and references therein.

Motivated by the above definition, we introduce the following higher anti-derivation.

Definition 1.3 Let $C$ be a commutative ring with unity, and $A$ be an algebra over $C$. Let $D = \{ \delta_{n}\}_{n\in \mathbb{N}}$ be a sequence of additive maps from $A$ into itself with $\delta_0 = id_A$. If

for all $a, b\in A$ and all $n\in\mathbb{N}$, then $D$ is called a higher anti-derivation. If $\delta_n$ is not necessarily additive, then $D$ is called a non-linear higher anti-derivable mapping.

Our main purpose in this paper is to show that every nonlinear higher anti-derivable mapping on a generalized matrix algebra is additive. In the following section, we introduce some basic concepts and the properties of generalized matrix algebras we require. Generalized matrix algebra is a particular structure of generalized $n$-matrix rings (see for example ^[21]), if we do not consider the scalar multiplication.

2.   Generalized matrix algebras

Let $\mathcal{R}$ be a commutative ring with identity, $\mathcal{A}$ and $\mathcal{B}$ be two unital $\mathcal{R}$-algebras, and $1_{\mathcal A}$ and $1_{\mathcal B}$ be the unit elements of $\mathcal{A}$ and $\mathcal{B}$ respectively. Let ${\mathcal M}$ be a faithful $({\mathcal A}, {\mathcal B})$-bimodule (i.e., for any $A\in\mathcal{A}, B\in\mathcal{B}$, if ${A\mathcal M} = 0$, then $A = 0$; if ${\mathcal M}B = 0$, then $B = 0$), and ${\mathcal N}$ be a (not necessarily faithful) $({\mathcal B}, {\mathcal A})$-bimodule. Suppose that there are two bimodule homomorphisms $\Phi_{\mathcal{MN}}: \mathcal{M}\otimes _\mathcal{B} \mathcal{N} \mapsto \mathcal{A}$ and $\Psi_{\mathcal{NM}}:\mathcal{N}\otimes_\mathcal{A} \mathcal{M}\mapsto \mathcal{B}$ satisfying the following associativity conditions: $(MN)M' = M(NM')$ and $(NM)N' = N(MN')$ for all $M, M'\in\mathcal{M}, N, N'\in\mathcal{N}$, where $MN = \Phi_{\mathcal{MN}}(M\otimes_\mathcal{B} N)$ and $NM = \Psi_{\mathcal{NM}}(N\otimes_\mathcal{A} M)$. Then

is an $\mathcal{R}$-algebra under the usual matrix-like addition, and the following multiplication:

for all $A, A^{'}\in\mathcal{A}, M, M^{'}\in\mathcal{M}, N, N^{'}\in\mathcal{N}$ and $B, B^{'}\in\mathcal{B}$, where at least one of the two bimodules $\mathcal{M}$ and $\mathcal{N}$ is distinct from zero. Such an $\mathcal{R}$-algebra is called a generalized matrix algebra. This type of algebra was first introduced by Morita ^[22]. In the following, we simply write $\mathcal{G}(\mathcal{A, M, N, B})$ as $\mathcal{G}$. For any associative algebra $\mathcal{A}$, if $\mathcal{A}$ is unital with the identity $1_{\mathcal{A}}$, and has a non-trivial idempotent $P$ ($P^{2} = P$, $P\neq0$ and $P\neq 1_{\mathcal{A}}$), then the Peirce decomposition of $\mathcal{A}$ corresponding to $P$ is $\mathcal{A} = P{\mathcal A}P+P{\mathcal A}Q+Q{\mathcal A}P+Q{\mathcal G}Q$, where $Q = 1_{\mathcal{A}}-P$. With respect to this decomposition, $\mathcal{A}$ is a generalized matrix algebra, and we then know that any an associative algebra containing a non-trivial idempotent is a generalized algebra.

Consider a generalized matrix algebra $\mathcal{G}$, let $1$ be the unit of $\mathcal{G}$. Set

and $\mathcal {G}_{ij} = P_{i}{\mathcal G}P_{j}(1\leq i, j\leq 2)$. Then, $\mathcal{G}$ can be represented as

where $\mathcal {G}_{11}$ is a subalgebra of $\mathcal {G}$ isomorphic to $\mathcal {A}$, $\mathcal {G}_{22}$ is a subalgebra of $\mathcal {G}$ isomorphic to $\mathcal {B}$, $\mathcal {G}_{12}$ is a $(\mathcal{G}_{11}, \mathcal{G}_{22})$-bimodule isomorphic to $\mathcal {M}$, and $\mathcal {G}_{21}$ is a $(\mathcal{G}_{22}, \mathcal{G}_{11})$-bimodule isomorphic to $\mathcal {N}$. Thus, $\mathcal {G}_{12}$ is a faithful $(\mathcal{G}_{11}, \mathcal{G}_{22})$-bimodule. Furthermore, for any $A\in \mathcal{G}$, $A$ can be represented as $A = A_{11}+A_{12}+A_{21}+A_{22}$, where $A_{ij}\in\mathcal {G}_{ij}$ $(1\leq i\leq j \leq 2)$.

In this section, our main result is the following Theorem 2.1. In ^[12], Daif proved every multiplicative derivation on a ring having an idempotent element which satisfies some conditions is additive. It is not hard to see an anti-derivation on a generalized matrix algebra $\mathcal{G}$ is a derivation from $\mathcal{G}$ into its anti-algebra. However, Theorem 2.1 is not a direct corollary of the theorem in ^[12]. This is because there are no idempotent elements in a generalized matrix algebra satisfying the conditions in ^[12]. Further, in ^[23], Ferreira and Sandhu showed that multiplicative anti-derivations are additive on generalized $n$-matrix rings. When $n = 2$, a generalized $n$-matrix ring is just the generalized matrix ring. However, these results are not the same as the following results.

Theorem 2.1 Let $\mathcal{G}$ be a generalized matrix algebra, and $\varphi$ be a mapping of $\mathcal{G}$ (without assumption of additivity). If $\varphi$ satisfies

for all $X, Y\in\mathcal{G}$, then $\varphi$ is additive.

In order to prove Theorem 2.1, we introduce Lemmas 2.1–2.4, and then prove that Lemmas 2.1–2.4 hold.

Lemma 2.1 If $\varphi$ is an nonlinear anti-derivable mapping on $\mathcal{G}$, then

(ⅰ) $\varphi(0) = 0$;

(ⅱ) $\varphi(P_{1}) = P_{1}\varphi(P_{1})P_{2}+P_{2}\varphi(P_{1})P_{1}$;

(ⅲ) $\varphi(P_{2}) = P_{1}\varphi(P_{2})P_{2}+P_{2}\varphi(P_{2})P_{1}$;

(ⅳ) $\varphi(P_{1}) = -\varphi(P_{2})$.

Proof (ⅰ) Taking $X = Y = 0$ in Eq (2.1), we have $\varphi(0) = \varphi(0)0+0\varphi(0) = 0$, and so $\varphi(0) = 0$.

(ⅱ) Taking $X = P_{1}, Y = P_{1}$ in Eq (2.1), we get $\varphi(P_{1}) = \varphi(P_{1})P_{1}+P_{1}\varphi(P_{1})$, which implies that $P_{1}\varphi(P_{1})P_{1} = P_{2}\varphi(P_{1})P_{2} = 0$. Hence, we obtain that $\varphi(P_{1}) = P_{1}\varphi(P_{1})P_{2}+P_{2}\varphi(P_{1})P_{1}$. Similarly, we can show (ⅲ) holds.

(ⅳ) Taking $X = P_{1}, Y = P_{2}$ in Eq (2.1), we get

Similarly, we get

Adding the above two equations, it follows from Lemma 2.1 (ⅱ) and (ⅲ) that

Therefore, $\varphi(P_{1}) = -\varphi(P_{2})$. The proof is completed.

Lemma 2.2 If $\varphi$ is an nonlinear anti-derivable mapping on $\mathcal{G}$, then for all $A_{11}\in \mathcal{G}_{11}, A_{12}\in \mathcal{G}_{12}, A_{21}\in \mathcal{G}_{21}, A_{22}\in \mathcal{G}_{22}$,

(ⅰ) $\varphi(A_{12}) = P_{2}\varphi(A_{12})P_{1}$;

(ⅱ) $\varphi(A_{21}) = P_{1}\varphi(A_{21})P_{2}$;

(ⅲ) $\varphi(A_{11}) = P_{1}\varphi(A_{11})P_{2}+P_{2}\varphi(A_{11})P_{1}$;

(ⅳ) $\varphi(A_{22}) = P_{1}\varphi(A_{22})P_{2}+P_{2}\varphi(A_{22})P_{1}$;

(ⅴ) $\varphi(P_{1})A_{12} = \varphi(P_{2})A_{12} = A_{12}\varphi(P_{1}) = A_{12}\varphi(P_{2}) = 0$;

(ⅵ) $\varphi(P_{1})A_{21} = \varphi(P_{2})A_{21} = A_{21}\varphi(P_{1}) = A_{21}\varphi(P_{2}) = 0$.

Proof (ⅰ) For any $A_{12}\in\mathcal{G}_{12}$, taking $X = P_{1}, Y = A_{12}$ in Eq (2.1), we have

This yields from $P_{2}\varphi(P_{1})P_{2} = 0$ that

Similarly, we have

This implies that

Therefore, we get $\varphi(A_{12}) = P_{2}d_{1}(A_{12})P_{1}$. Similarly, we can show that (ⅱ) holds.

(ⅲ) For any $A_{11}\in \mathcal{G}_{11}, A_{12}\in\mathcal{G}_{12}$, taking $X = A_{12}, Y = A_{11}$ in Eq (2.1), then by Lemma 2.2 (ⅰ), we have

This yields from the faithfulness of ${\mathcal G}_{12}$ that

Similarly, taking $X = P_{2}, Y = A_{11}$ in Eq (2.1), we have

This implies that

Therefore, we obtain that $\varphi(A_{11}) = P_{1}\varphi(A_{11})P_{2}+P_{2}\varphi(A_{11})P_{1}$. Similarly, we can show that (ⅳ) holds.

(ⅴ) For any $A_{12}\in\mathcal{G}_{12}$, it follows from Eqs (2.2)–(2.3) and $\varphi(A_{12}) = P_{2}\varphi(A_{12})P_{1}$ that

Therefore, we obtain from $\varphi(P_{1}) = -\varphi(P_{2})$ that $\varphi(P_{1})A_{12} = \varphi(P_{2})A_{12} = A_{12}\varphi(P_{1}) = A_{12}\varphi(P_{2}) = 0$. Similarly, we can show that (ⅵ) holds.

Lemma 2.3 If $\varphi$ is a nonlinear anti-derivable mapping on $\mathcal{G}$, then for all $A_{11}, B_{11}\in \mathcal{G}_{11}, A_{12}, B_{12}\in \mathcal{G}_{12}, A_{21}, B_{21}\in \mathcal{G}_{21}, A_{22}, B_{22}\in \mathcal{G}_{22}$,

(ⅰ) $\varphi(A_{11}+B_{11}) = \varphi(A_{11})+\varphi(B_{11})$;

(ⅱ) $\varphi(A_{22}+B_{22}) = \varphi(A_{22})+\varphi(B_{22})$;

(ⅲ) $\varphi(A_{11}+A_{12}) = \varphi(A_{11})+\varphi(A_{12})$;

(ⅳ) $\varphi(A_{12}+A_{22}) = \varphi(A_{12})+\varphi(A_{22})$;

(ⅴ) $\varphi(A_{21}+A_{22}) = \varphi(A_{21})+\varphi(A_{22})$;

(ⅵ) $\varphi(A_{12}+B_{12}) = \varphi(A_{12})+\varphi(B_{12})$;

(ⅶ) $\varphi(A_{21}+B_{21}) = \varphi(A_{21})+\varphi(B_{21})$.

Proof (ⅰ) For any $A_{11}, B_{11}\in \mathcal{G}_{11}$, taking $X = A_{11}, Y = P_{1}$ in Eq (2.1), we have

Taking $X = P_{2}, Y = A_{11}$ in Eq (2.1), we have $0 = \varphi(P_{2}A_{11}) = \varphi(A_{11})P_{2}+A_{11}\varphi(P_{2})$, this yields from $\varphi(P_{1}) = -\varphi(P_{2})$ that

Hence, we can get from above two equations and $\varphi(A_{11}) = P_{1}\varphi(A_{11})P_{2}+P_{2}\varphi(A_{11})P_{1}$ that

Similarly, we get

And,

Therefore, it follows from above three equations that $\varphi(A_{11}+B_{11}) = \varphi(A_{11})+\varphi(B_{11})$. Similarly, we can show (ⅱ) holds.

(ⅲ) For any $A_{11}\in \mathcal{G}_{11}, A_{12}\in \mathcal{G}_{12}$, taking $X = A_{11}+A_{12}, Y = P_{1}$ in Eq (2.1), we get that

Similarly, taking $X = A_{11}+A_{12}, Y = P_{2}$ in Eq (2.1), by Lemma 2.2 (ⅴ), we have

Adding the above two equations, we then obtain from $\varphi(P_{1}) = -\varphi(P_{2})$ that $\varphi(A_{11}+A_{12}) = \varphi(A_{11})+\varphi(A_{12})$. Similarly, we can show (ⅳ) and (ⅴ) hold.

(ⅵ) For any $A_{12}, B_{12}\in \mathcal{G}_{12}$, taking $X = A_{12}, Y = B_{12}$ in Eq (2.1), it follows from $A_{12}B_{12} = 0$ that

Since $A_{12}+B_{12} = (P_{1}+A_{12})(P_{2}+B_{12})$, we take $X = P_{1}+A_{12}, Y = P_{2}+B_{12}$ in Eq (2.1), and then we get from Lemma 2.2, Lemma 2.3(ⅰ), (ⅴ), Lemma 2.4(ⅲ)-(ⅳ) and Eq (2.4) that

Similarly, we can show (vii) holds. The proof is completed.

Lemma 2.4 If $\varphi$ is a nonlinear anti-derivable mapping on $\mathcal{G}$, then $\varphi(A_{11}+A_{12}+A_{21}+A_{22}) = \varphi(A_{11})+\varphi(A_{12})+\varphi(A_{21})+\varphi(A_{22})$ for all $A_{11}\in \mathcal{G}_{11}, A_{12}\in \mathcal{G}_{12}, A_{21}\in \mathcal{G}_{21}$ and $A_{22}\in \mathcal{G}_{22}$.

Proof For any $A_{11}\in \mathcal{G}_{11}, A_{12}\in \mathcal{G}_{12}, A_{21}\in \mathcal{G}_{21}, A_{22}\in \mathcal{G}_{22}$, taking $X = P_{1}, Y = A_{11}+A_{12}+A_{21}+A_{22}$ in Eq (2.1), we get from Lemma 2.3 (ⅲ), (ⅴ) and Lemma 2.2(ⅴ)-(ⅵ) that

Similarly, we obtain that

Adding the above two equations, using $\varphi(P_{1}) = -\varphi(P_{2})$, we get $\varphi(A_{11}+A_{12}+A_{21}+A_{22}) = \varphi(A_{11})+\varphi(A_{12})+\varphi(A_{21})+\varphi(A_{22})$. The proof is completed.

Now, we complete the proof of Theorem 2.1.

Proof of Theorem 2.1 For any $X, Y\in\mathcal{G}$, set $X = A_{11}+A_{12}+A_{21}+A_{22}$ and $Y = B_{11}+B_{12}+B_{21}+B_{22}$, where $A_{ij}, B_{ij}\in \mathcal{G}_{ij}(1 \leq i\leq j \leq 2)$, then by Lemmas 2.3 and 2.4, we obtain that

Therefore, $\varphi$ is an additive mapping on $\mathcal{G}$. The proof is completed.

Next, we will give the second main result.

3.   Additivity of nonlinear higher anti-derivable mappings on generalized matrix algebras

Theorem 3.1 Let $\mathcal{G}$ be a generalized matrix algebra and $D = \{d_{n}\}_{n\in \mathbb{N}}$ be a sequence mapping from $\mathcal{G}$ into itself (without assumption of additivity) such that

for any $n\in \mathbb{N}, X, Y\in\mathcal{G}$, then $D$ is an additive mapping on $\mathcal{G}$.

In the following, to prove Theorem 3.1, we will introduce Lemmas 3.1–3.3, and then use mathematical induction to prove that Lemmas 3.1–3.3 hold. We assume that $\mathcal{G}$ is a generalized matrix algebra, and $D = \{d_{n}\}_{n\in \mathbb{N}}$ is a higher anti-derivable mapping on $\mathcal{G}$. Let ${\mathbb{N}}$ be the set of non-negative integers, ${\mathbb{N}^{+}}$ be the set of positive integers, and $i, j, k, p, q, n\in\mathbb{N}$. For any $X, Y\in\mathcal{G}$, $A_{11}\in \mathcal{G}_{11}, A_{12}\in \mathcal{G}_{12}, A_{21}\in \mathcal{G}_{21}, A_{22}\in \mathcal{G}_{22}$. We say a map $f:\mathcal{G}\to\mathcal{G}$ satisfies the set of properties $\mathcal{L}$, if

(ⅰ) $f(X+Y) = f(X)+f(Y)$;

(ⅱ) $f(0) = 0, f(P_{1}) = -f(P_{2})\in\mathcal{M}+\mathcal{N}$;

(ⅲ) $f(A_{12}) = P_{2}f(A_{12})P_{1}$;

(ⅳ) $f(P_{1})A_{12} = f(P_{2})A_{12} = A_{12}f(P_{1}) = A_{12}f(P_{2}) = 0$;

(ⅴ) $f(A_{21}) = P_{1}f(A_{21})P_{2}$;

(ⅵ) $f(P_{1})A_{21} = f(P_{2})A_{21} = A_{21}f(P_{1}) = A_{21}f(P_{2}) = 0$;

(ⅶ) $f(A_{11}) = P_{1}f(A_{11})P_{2}+P_{2}f(A_{11})P_{1}$;

(ⅷ) $f(A_{22}) = P_{1}f(A_{22})P_{2}+P_{2}f(A_{22})P_{1}$.

It is known from Theorem 2.1 that $d_{1}$ satisfies the set of properties $\mathcal{L}$. Now, for any $X, Y\in\mathcal{G}$, $A_{11}\in \mathcal{G}_{11}, A_{12}\in \mathcal{G}_{12}, A_{21}\in \mathcal{G}_{21}, A_{22}\in \mathcal{G}_{22}$, we assume that $d_{k}$($1\leqslant k < n$) satisfies the set of properties $\mathcal{L}$. In the following, we show $d_n$ satisfies the set of properties $\mathcal{L}$.

Lemma 3.1 For any $n\in\mathbb{N}^{+}, A_{11}\in \mathcal{G}_{11}, A_{12}\in \mathcal{G}_{12}, A_{21}\in \mathcal{G}_{21}, A_{22}\in \mathcal{G}_{22}$, $d_n$ satisfies the set of properties $\mathcal{L}$.

Proof (ⅰ) For any $n\in\mathbb{N}^{+}$, taking $X = Y = 0$ in Eq (3.1), it follows from the set of properties $\mathcal{L}$(ⅱ) that

For any $n, i, j\in\mathbb{N}^{+} (i, j < n)$, since $d_{i}(P_{1}), d_{j}(P_{1}), d_{i}(P_{2}), d_{j}(P_{2})\in\mathcal{M}+\mathcal{N}$, and so by the set of properties $\mathcal{L}$ (ⅳ) and (ⅵ), we get that

Taking $X = P_{1}, Y = P_{1}$ in Eq (3.1), by Eq (3.2), we get

This implies that

Similarly, we have

Taking $X = P_{1}, Y = P_{2}$ in Eq (3.1), by Eq (3.2), we get

Therefore, we get

Similarly, we obtain that

Therefore, by Eqs (3.3)–(3.6), we get that $d_{n}(P_{1}) = -d_{n}(P_{2})\in\mathcal{M}+\mathcal{N}$.

(ⅱ)-(ⅲ) For any $n\in\mathbb{N}^{+}$, $A_{12}\in\mathcal{G}_{12}$, taking $X = P_{1}, Y = A_{12}$ in Eq (3.1), it follows from (ⅲ) and (ⅵ) of the set of properties $\mathcal{L}$ that

This yields from $P_{2}d_{n}(P_{1})P_{2} = 0$ that

Similarly, we get

This yields that

Therefore, by $d_{n}(P_{1}) = -d_{n}(P_{2})$ and Eqs (3.7) and (3.8), we get (ⅱ) and (ⅲ). Similarly, we can show that (ⅳ) and (ⅴ) hold.

(ⅵ) For any $n\in\mathbb{N}^{+}, A_{11}\in\mathcal{G}_{11}, A_{12}\in\mathcal{G}_{12}$, taking $X = A_{11}, Y = A_{12}$ in Eq (3.1), it follows from the set of properties $\mathcal{L}$ (ⅱ) and Lemma 3.1 (ⅱ) that

This implies that $P_{1}d_{n}(A_{11})P_{1}A_{12} = 0$, and so by the faithfulness of ${\mathcal G}_{12}$, we get

Taking $X = A_{11}, Y = P_{1}$ in Eq (3.1), we get from (ⅳ), (ⅵ) and (vii) of the set of properties $\mathcal{L}$ and Lemma 3.1 (vii) that

This yields that

Similarly, taking $X = P_{2}, Y = A_{11}$ in Eq (3.1), we get from (ⅳ), (ⅵ) and (vii) of the set of properties $\mathcal{L}$ and Lemma 3.1 (vii) that

This yields that

Therefore, we get from Eqs (3.9)–(3.11) that

Similarly, we can show that (viii) holds. The proof is completed.

Lemma 3.2 For any $n\in\mathbb{N}^{+}$, $A_{11}, B_{11}\in \mathcal{G}_{11}, A_{12}, B_{12}\in \mathcal{G}_{12}, A_{21}, B_{21}\in \mathcal{G}_{21}, A_{22}, B_{22}\in \mathcal{G}_{22}$, then

(ⅰ) $d_{n}(A_{11}+B_{11}) = d_{n}(A_{11})+d_{n}(B_{11})$;

(ⅱ) $d_{n}(A_{22}+B_{22}) = d_{n}(A_{22})+d_{n}(B_{22})$;

(ⅲ) $d_{n}(A_{11}+A_{12}) = d_{n}(A_{11})+d_{n}(A_{12})$;

(ⅳ) $d_{n}(A_{12}+A_{22}) = d_{n}(A_{12})+d_{n}(A_{22})$;

(ⅴ) $d_{n}(A_{21}+A_{22}) = d_{n}(A_{21})+d_{n}(A_{22})$;

(ⅵ) $d_{n}(A_{12}+B_{12}) = d_{n}(A_{12})+d_{n}(B_{12})$;

(vii) $d_{n}(A_{21}+B_{21}) = d_{n}(A_{21})+d_{n}(B_{21})$.

Proof (ⅰ) For any $n\in\mathbb{N}^{+}, A_{11}, B_{11}\in \mathcal{G}_{11}$, we get from Eq (3.12) that

Similarly, we show that (ⅱ) holds.

(ⅲ) For any $n\in\mathbb{N}^{+}, A_{11}\in \mathcal{G}_{11}, A_{12}\in \mathcal{G}_{12}$, taking $X = A_{11}+A_{12}, Y = P_{1}$ in Eq (3.11), we get from the set of properties $\mathcal{L}$ (ⅰ) and Lemma 3.1 that

Thus, we get

Similarly, taking $X = A_{11}+A_{12}, Y = P_{2}$ in Eq (3.1), we obtain that

Adding the above two equations, we obtain from $d_{n}(P_{1}) = -d_{n}(P_{2})$ that $d_{n}(A_{11}+A_{12}) = d_{n}(A_{11})+d_{n}(A_{12})$. Similarly, we can show (ⅳ) and (ⅴ) hold.

(ⅵ) For any $A_{12}, B_{12}\in \mathcal{G}_{12}$, taking $X = A_{12}, Y = B_{12}$ in Eq (3.1), then it follows from $A_{12}B_{12} = 0$ that

Since $A_{12}+B_{12} = (P_{1}+A_{12})(P_{2}+B_{12})$, we take $X = P_{1}+A_{12}, Y = P_{2}+B_{12}$ in Eq (3.1), and then we get from Lemma 3.1, Lemma 3.2(ⅲ)-(ⅳ), and Eq (3.13) that

Similarly, we can show (vii) holds. The proof is completed.

Lemma 3.3 For any $n\in\mathbb{N}^{+}$, $A_{11}\in \mathcal{G}_{11}, A_{12}\in \mathcal{G}_{12}, A_{21}\in \mathcal{G}_{21}$ and $A_{22}\in \mathcal{G}_{22}$, then $d_{n}(A_{11}+A_{12}+A_{21}+A_{22}) = d_{n}(A_{11})+d_{n}(A_{12})+d_{n}(A_{21})+d_{n}(A_{22})$.

Proof For any $n\in\mathbb{N}^{+}$, $A_{11}\in \mathcal{G}_{11}, A_{12}\in \mathcal{G}_{12}, A_{21}\in \mathcal{G}_{21}, A_{22}\in \mathcal{G}_{22}$, taking $X = P_{1}, Y = A_{11}+A_{12}+A_{21}+A_{22}$ in Eq (3.1), we obtain from the set of properties $\mathcal{L}$ (ⅰ), Lemma 3.2 (ⅲ) and (ⅴ) that

Similarly, we take $X = P_{2}, Y = A_{11}+A_{12}+A_{21}+A_{22}$ in Eq (3.1), then we obtain that

Adding the above two equations, by $d_{n}(P_{1})+d_{n}(P_{2}) = 0$, we get that $d_{n}(A_{11}+A_{12}+A_{21}+A_{22}) = d_{n}(A_{11})+d_{n}(A_{12})+d_{n}(A_{21})+d_{n}(A_{22})$. The proof is completed.

Now, we complete the proof of Theorem 3.1.

Proof of Theorem 3.1 For any $n\in\mathbb{N}^{+}$, $X, Y\in\mathcal{G}$, set $X = A_{11}+A_{12}+A_{21}+A_{22}$ and $Y = B_{11}+B_{12}+B_{21}+B_{22}$, where $A_{ij}, B_{ij}\in \mathcal{G}_{ij}(1 \leq i\leq j \leq 2)$, then, by Lemmas 3.2 and 3.3, we can obtain that

Therefore, $D = \{d_{n}\}_{n\in \mathbb{N}}$ is an additive mapping on $\mathcal{G}$. The proof is completed.

In the following, we give some applications of Theorem 3.1.

Because triangular algebras and full matrix algebras are two special classes of generalized matrix algebras, we can get Corollaries 3.1–3.5 immediately. For the definition of triangular algebra, we refer readers to ^[24]. It is worth pointing out that, in ^[25], Ferreira showed that under certain conditions, every $m$-multiplicative derivation on a triangular $n$-matrix ring is additive.

Corollary 3.1 Let ${\mathcal A}$ and ${\mathcal B}$ be unital algebras, ${\mathcal M}$ be a unital $({\mathcal A}, {\mathcal B})$-bimodule, which is faithful as both a left ${\mathcal A}$-module and a right ${\mathcal B}$-module, and $\mathcal{U} = \mathrm{Tri}({\mathcal A}, {\mathcal M}, {\mathcal B}) = \left\{\left(\begin{array} {ccc}a&m\\0&b\\\end{array}\right): a\in{\mathcal A}, m\in{\mathcal M}, b\in{\mathcal B} \right\}$ be a triangular algebra. If $D = \{d_{n}\}_{n\in \mathbb{N}}$ is a nonlinear higher anti-derivable mapping on $\mathcal {U}$, then $D = \{d_{n}\}_{n\in \mathbb{N}}$ is additive.

Corollary 3.2 Let ${\mathcal A}$ be an unital algebra, and $\mathcal{M}_{n}(\mathcal{A}) (2\leq n)$ be the full matrix algebras of all $n\times n$ matrices over ${\mathcal A}$. If $D = \{d_{n}\}_{n\in \mathbb{N}}$ is a nonlinear higher anti-derivable mapping on $\mathcal{M}_{n}(\mathcal{A})$, then $D = \{d_{n}\}_{n\in \mathbb{N}}$ is additive.

Corollary 3.3 Let $\mathcal{R}$ be a unital prime ring with a nontrivial idempotent $P$, and $I$ be the unit of $\mathcal{R}$. If $D = \{d_{n}\}_{n\in \mathbb{N}}$ is a nonlinear higher anti-derivable mapping on $\mathcal {U}$, then $D = \{d_{n}\}_{n\in \mathbb{N}}$ is additive.

Proof of Corollary 3.3 Suppose $Q = I-P$. Since $\mathcal{R}$ a prime ring, it follows that $P\mathcal{R}Q$ is a faithful ($P\mathcal{R}P, Q\mathcal{R}Q$)-bimodule. Then, $\mathcal{R}$ is isomorphic to the generalized matrix algebra

Therefore, by Theorem 3.1, we know that $D = \{d_{n}\}_{n\in \mathbb{N}}$ is additive.

Since standard operator algebras and factor von Neumann algebras are prime algebras with nontrivial idempotents, by Corollary 3.3, we obtain Corollary 3.4 and Corollary 3.5 as follows.

Corollary 3.4 Let $\mathcal{X}$ be a Banach space over number field $\mathcal {F}$, and $\mathcal{A}(\mathcal{X})$ be an unital standard operator algebra over $\mathcal{X}$. If $D = \{d_{n}\}_{n\in \mathbb{N}}$ is a nonlinear higher anti-derivable mapping on $\mathcal{A}(\mathcal{X})$, then $D = \{d_{n}\}_{n\in \mathbb{N}}$ is additive.

Corollary 3.5 Let $\mathcal{H}$ be a Hilbert space over number field $\mathcal {F}$, and $\mathcal V$ be a factor von Neumann algebra over $\mathcal{H}$. If $D = \{d_{n}\}_{n\in \mathbb{N}}$ is a nonlinear higher anti-derivable mapping on $\mathcal{V}$, then $D = \{d_{n}\}_{n\in \mathbb{N}}$ is additive.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work was supported by Talent Project Foundation of Yunnan Province (No.202105AC160089), Natural Science Foundation of Yunnan Province(No.202101BA070001198), and Basic Research Foundation of Yunnan Education Department(Nos.2020J0748, 2021J0635).

Conflict of interest

The authors declare there is no conflicts of interest.