Nonlinear Jordan Derivable Mappings of Generalized Matrix Algebras by Lie Product Square-Zero Elements

-e aim of the paper is to give a description of nonlinear Jordan derivable mappings of a certain class of generalized matrix algebras by Lie product square-zero elements. We prove that under certain conditions, a nonlinear Jordan derivable mapping Δ of a generalized matrix algebra by Lie product square-zero elements is a sum of an additive derivation δ and an additive antiderivation f. Moreover, δ and f are uniquely determined.


Introduction
roughout the paper, by an algebra we shall mean an algebra over a fixed unital commutative ring R. Let A be a unital algebra, P be a fixed element of A, Ω � A ∈ A: A 2 � 0 , and Δ: A⟼A be an additive (resp., without assumption of additivity) mapping. For any A, B ∈ A, denote the Jordan product (resp., Lie product) of A and B by A ∘ B � AB + BA (resp., [A, B] � AB − BA). For any A ∈ A, if 2A � 0, implying A � 0, then A is said to be a 2-torsion free algebra. Recall that Δ is called an additive derivation (resp., nonlinear derivable mapping) if Δ(AB) � for all A, B ∈ A with [A, B] ∈ Ω, then we say Δ is a nonlinear Jordan derivable mapping of A by Lie product square-zero elements. Obviously, every additive derivation or additive antiderivation is an additive Jordan derivation. However, the inverse statement is not true in general (see [1]). A natural and very interesting problem that we are dealing with is studying those conditions on a ring or an algebra such that every additive Jordan derivation or every nonlinear Jordan derivable map is an additive derivation.
In the past few decades, the research on this problem has attracted the attention of many mathematicians. For examples, Herstein in [2] proved that every additive Jordan derivation on a prime ring not of characteristic 2 is an additive derivation. Later, this result was extended by Cusack in [3] and Brešar and Vukman in [4] to the case of semiprime ring, respectively. Zhang and Yu in [5,6] showed that every additive Jordan derivation on a nest algebra and a 2-torsion free triangular algebra is an inner derivation and an additive derivation, respectively. For some conclusions about P point Jordan derivable mapping, we refer the readers to [7][8][9] and references therein for more details. Lu in [10] showed that every nonlinear Jordan derivable mapping on a 2-torsion free semiprime ring is an additive derivation. Ashraf and Jabeen in [11] showed that every nonlinear Jordan triple derivable mapping on a 2-torsion free triangular algebra is an additive derivation. In particular, Benkovič in [1] proved that every additive Jordan derivation from an upper triangular matrix algebra to its bimodule is a sum of an additive derivation and an additive antiderivation. Li et al. in [12] proved that under certain conditions every additive Jordan derivation on a generalized matrix algebra is a sum of an additive derivation and an additive antiderivation. For other similar results, we refer the readers to [13,14] and references therein for more details.
Inspired by the above references, in this paper, we study the nonlinear Jordan derivable mapping of generalized matrix algebras by Lie product square-zero elements, we get that under certain conditions, a nonlinear Jordan derivable mapping of a generalized matrix algebra by Lie product square-zero elements is a sum of an additive derivation and an additive antiderivation.

Generalized Matrix Algebras
Let A and B be two unital algebras with unit elements 1 A and 1 B , respectively, M be a faithful (A, B)-bimodule, and N be a (B, A)-bimodule. en, is a R-algebra under matrix-like addition and multiplication, satisfying the following commutative diagrams: where Φ MN : M ⊗ B N⟼A and Ψ NM : N ⊗ A M⟼B are two bimodule homomorphisms. Such a R-algebra G is called a generalized matrix algebra. For convenience in reading, we give the multiplication of generalized matrix algebra as follows: for all A, A ′ ∈ A, M, M ′ ∈ M, N, N ′ ∈ N, and B, B ′ ∈ B. Furthermore, when N � 0, then G is a triangular algebra. e most common examples of generalized matrix algebras are triangular algebras and full matrix algebras (see [15,16] for details).
Considering a generalized matrix algebra G, let 1 be the unit element of G. Set and Hence, for any A ∈ G, A can be represented as

Main Results
In this paper, our main result is the following theorem. In order to prove eorem 1, we introduce some lemmas. Next, we assume that G is a 2-torsion generalized matrix algebra, M is a faithful (A, B)-bimodule and N is a (B, A)-bimodule (i.e., G 12 is a faithful (G 11 , G 22 )-bimodule and G 21 is a (G 22 , G 11 )-bimodule), MN � NM � 0 (i.e., Ω � A ∈ G: A 2 � 0 , and Δ is a nonlinear Jordan derivable mapping of G by Lie product square-zero elements.

Lemma 1. For any
Proof (i) Since Δ is a nonlinear Jordan derivable mapping of G by Lie product square-zero elements, then for any Multiplying equation (9) from both the sides by P j and then by the property of 2-torsion freeness of G, we have Multiplying equation (9) from the left by P i and from the right by P j , we get Next, we show that P i Δ(P i )P i � 0 (1 ≤ i ≤ 2). Indeed, for any X 12 ∈ G 12 , since [P 1 , X 12 ] � X 12 ∈ Ω and taking A � P 1 and B � X 12 in equation (8), we obtain Multiplying equation (12) from the left by P 1 and from the right by P 2 and then by equation (10), we have P 1 Δ(P 1 )X 12 � P 1 Δ(P 1 )P 1 X 12 � 0; similarly, we can get that X 12 Δ(P 2 )P 2 � X 12 P 2 Δ(P 2 )P 2 � 0; therefore, we obtain from the faithfulness of G 12 that us, we obtain from equations (10)-(12) that Multiplying (14) by P i and P j from both sides, respectively, we get that Multiplying equation (16) by P j from both sides and then by the property of 2-torsion freeness of G, we have Multiplying equation (16) from the left by P i and from the right by P j , it follows from Lemma 1 (ii) that Similarly, we can show that P j Δ(A ii )P i � Δ(P i )A ii holds. e proof is completed. (8), and so by Lemma 1 (ii)-(iv), we have

Lemma 2. For any
is yields that Multiplying equation (19) by P j from both sides and then by the property of 2-torsion freeness of G and Lemma 1 (iii), we have Multiplying equation (19) from the left by P i and from the right by P j , we get Multiplying equation (19) from the left by P j and from the right by P i , we get Next, we show that Indeed, for any A 11 ∈ G 11 , A 12 , X 12 ∈ G 12 , since [A 11 + A 12 , X 12 ] � [A 11 , X 12 ] � A 11 X 12 ∈ Ω, on the one hand, taking A � A 11 + A 12 , B � X 12 in equation (8), and then by Lemma 1 (iii), we get On the other hand, taking A � A 11 and B � X 12 in equation (8), we get that

Journal of Mathematics
Comparing equations (23) and (24), we get Multiplying equation (25) from the left by P 1 and from the right by P 2 , then by equation (20) and Lemma 1 (iv), we get Similarly, for any X 12 ∈ G 12 , A 21 ∈ G 21 , and A 22 ∈ G 22 , we can get that is yields from the faithfulness of G 12 that erefore, it follows from equations (20)-(28) and (8), and so by Lemma 2 (i)-(ii) and Lemma 1 (ii)-(iv), we have taking A � P j + A ij andB � A ji + P i in equation (8) and we get from MN � NM � 0, Lemma 2 (i)-(ii), and Lemma 1 (ii)-(iv) that Journal of Mathematics (v) For any A 11 , B 11 ∈ G 11 , it follows from Lemma 1 (iv) that In the following, we show that P 1 Δ(A 11 + B 11 )P 1 � P 1 Δ(A 11 )P 1 + P 1 Δ(B 11 )P 1 . Indeed, for any A 11 , B 11 ∈ G 11 , X 12 ∈ G 12 , on the one hand, since [A 11 + B 11 , X 12 ] � (A 11 + B 11 ) X 12 ∈ Ω and taking A � A 11 + B 11 and B � X 12 in equation (8), then we get On the other hand, since [A 11 , X 12 ] � A 11 X 12 ∈ Ω and [B 11 , X 12 ] � B 11 X 12 ∈ Ω, we get, respectively, erefore, it follows from Lemma 2 (iii) and equations (34)-(36) that Multiplying equation (37) from the left by P 1 and from the right by P 2 and then by Lemma 1 (iv), we get erefore, by the faithfulness of G 12 , we get that Hence, we get from equations (31)-(33) and (39) and Lemma 1 (iv) that Δ(A 11 + B 11 ) � Δ(A 11 ) + Δ(B 11 ). Similarly, we can show that (8) and then by Lemma 1 (ii)-(iv) and Lemma 2 (iv), we obtain 6 Journal of Mathematics is implies that In the following, we show that (8) and so by MN � NM � 0 and Lemma 1 (iii), we have On the other hand, since [A 11 , X 12 ] � A 11 X 12 ∈ Ω, we get Comparing equations (44) and (45), we get Multiplying equation (46) from the left by P 1 and from the right by P 2 and then by Lemma 1 (iv), equation (43), and the faithfulness of G 12 , we get Similarly, for any A 22 ∈ G 22 , A 12 ∈ G 12 , and A 21 ∈ G 21 , we can get that

Hence, for any
erefore, it follows from equations (41)-(48) and taking A � A ii + A ij + A ji + A jj , B � P j in equation (8) and so by Lemma 2 (vi) and Lemma 1 (iv), we have Similarly, we obtain erefore, it follows from (50) and (51), Lemma 1 (ii), and the property of 2-torsion freeness of G that Δ( e proof is completed. (52) erefore, Δ is an additive mapping on G. e proof is completed. □ Remark 1. For any A ∈ G, we define a mapping f: G⟼G as en, by the definition of f, we can easily obtain that Next, we will show that f is an additive antiderivation on G. First, we introduce Lemma 4 and get that f is an additive mapping, and then, we introduce Lemmas 5 and 6 and show that f is an additive antiderivation on G. Proof. For any A, B ∈ G, since we have shown that Δ is an additive mapping in Lemma 3 and then by the definition of f, we obtain that 8 Journal of Mathematics (55) erefore, f is an additive mapping on G. e proof is completed.

Lemma 5. Let f: G⟼G be as in Remark 1. en, for any
(v) Similar to (iv), we can show that (v) holds. (vi) Similar to (iv), we can show that (vi) holds.
Hence, by the definition of f, we have . Similarly, we can show (viii) holds. e proof is completed. Proof. For any A, B ∈ G, let A � A 11 + A 12 + A 21 + A 22 and B � B 11 + B 12 + B 21 + B 22 , where A ij , B ij ∈ G ij (1 ≤ i, j ≤ 2), and we obtain from Lemmas 4 and 5 that Journal of Mathematics 9 erefore, f is an additive antiderivation on G. e proof is completed. □ Remark 2. For any A ∈ G, we define a mapping δ: G⟼G as en, we obtain from Lemmas 3 and 4 that δ is an additive mapping on G.
In the following, we will introduce Lemmas 7-9 and show that δ is an additive derivation on G.

Lemma 7. Let δ: G⟼G be as in Remark 2. en, for any
Proof. By Lemma 1 and Remarks 1 and 2, we can easily check that Lemma 7 holds. e proof is completed.

Lemma 8. Let δ: G⟼G be as in Remark 2. en, for any
And then, it follows from MN � NM � 0, Lemma 1, and Remark 1 that On the other hand, we obtain from Lemma 6 and erefore, by Remark 2, equations (61) and (62), (ii) Similarly, we can show that (ii) holds. (iii) For any A 11 , B 11 ∈ G 11 and Y 12 ∈ G 12 , by Lemma 8 (i), on the one hand, we get On the other hand, we have Comparing equations (64) and (65), we get is yields from the faithfulness of G 12 that Furthermore, by Lemma 7 (i) and (iii), we have erefore, we obtain from Lemma 7 (iii) and equations (67) and (68) that (iv) For any A ii ∈ G ii and B jj ∈ G jj (1 ≤ i ≠ j ≤ 2), by Lemma 1 (iv), we have And so, this yields from equation (70) that erefore, it follows from equation (72) and (v) For any A ij ∈ G ij and B ji ∈ G ji (1 ≤ i ≠ j ≤ 2), it follows from MN � NM � 0 and Lemma 7 (ii) that And, us, we obtain from (74) and (75) that (vi) Similar to (v), we can show that (vi) holds. (vii) For any A ii ∈ G ii , A ij ∈ G ij , and B ji ∈ G ji (1 ≤ i ≠ j ≤ 2), on the one hand, we get On the other hand, it follows from MN � NM � 0 and Lemma 7 (ii)-(iii) that Comparing equations (77) and (78), we get erefore, δ is an additive derivation on G. e proof is completed.
; by Remarks 1 and 2 and Lemmas 6 and 9, we obtain that where δ is an additive derivation and f is an additive antiderivation, respectively. Furthermore, f(A ii ) � 0 for all In the following, we check that δ and f are unique. Let d: G⟼G be an additive derivation and h: G⟼G be an additive antiderivation such that h(A ii ) � 0 for all A ii ∈ G ii (1 ≤ i ≤ 2). Suppose that en, for any A ∈ G, by eorem 1, we get that δ(A) for all A ∈ G; this yields from δ and d two additive derivations that h − f is an additive derivation; moreover, by . Indeed, for any A ij ∈ G ij (1 ≤ i ≠ j ≤ 2), on the one hand, since h − f is an additive derivation and h(P i ) � f(P i ) � 0(1 ≤ i ≤ 2) and so we get that On the other hand, since h and f are two additive antiderivations and h(P j ) � f(P j ) � 0(1 ≤ j ≤ 2), we get that □ Remark 3. By the above lemmas, Remarks 1 and 2, and the proof of eorem 1, we can easily obtain that if Δ is a nonlinear Jordan derivable mapping of G by Lie product square-zero elements, then the following statements are equivalent: Next, we give an application of Remark 3 to triangular algebras and we obtain that every nonlinear Jordan derivable mapping of a triangular algebra by Lie product square-zero elements is an additive derivation. : a ∈ A, m ∈ M, b ∈ B be a 2-torsion free triangular algebra, and Δ: U⟼U be a nonlinear Jordan derivable mapping of U by Lie product square-zero elements; then, Δ is an additive derivation.
Proof. of Corollary 1. Let 1 A and 1 B be the identities of the algebras A and B, respectively, and let 1 be the identity of the triangular algebra U. We denote It is clear that the triangular algebra U may be represented as U � P 1 UP 1 + P 1 UP 2 + P 2 UP 2 � A + M + B. (87) For any A ∈ U, let A � A 11 + A 12 + A 22 , where A ij ∈ U ij (1 ≤ i, j ≤ 2); then for any A 12 ∈ U 12 , since U 21 � 0 { } and by Lemma 1 (iii), we get Δ A 12 � P 1 Δ A 12 P 2 + P 2 Δ A 12 P 1 � P 1 Δ A 12 P 2 .
(88) erefore, we obtain from Remark 3 (iii) that Δ is an additive derivation. e proof is completed.

Data Availability
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