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

Fei, Xiuhai; Zhang, Haifang

doi:https://doi.org/10.1155/2021/2065425

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Classical Problems in Algebra and Number Theory

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Volume 2021 | Article ID 2065425 | https://doi.org/10.1155/2021/2065425

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

Xiuhai Fei¹and Haifang Zhang¹

Academic Editor: Wenpeng Zhang

Received30 Jul 2021

Accepted18 Aug 2021

Published10 Sept 2021

Abstract

The 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 of a generalized matrix algebra by Lie product square-zero elements is a sum of an additive derivation and an additive antiderivation . Moreover, and are uniquely determined.

1. Introduction

Throughout the paper, by an algebra we shall mean an algebra over a fixed unital commutative ring . Let be a unital algebra, be a fixed element of , , and be an additive (resp., without assumption of additivity) mapping. For any , denote the Jordan product (resp., Lie product) of by (resp., ). For any , if , implying , then is said to be a 2-torsion free algebra. Recall that is called an additive derivation (resp., nonlinear derivable mapping) if for all . It is called an additive antiderivation (resp., nonlinear antiderivable mapping) if for all . It is called an additive Jordan derivation (resp., nonlinear Jordan derivable mapping) if for all . It is called a point derivable mapping (resp., point nonlinear derivable mapping) if for all with . Similarly, we have the definitions of point Jordan derivable mapping and point nonlinear Jordan derivable mapping. In this paper, if is without assumption of additivity and satisfiesfor all with , then we say is a nonlinear Jordan derivable mapping of 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 point Jordan derivable mapping, we refer the readers to [7–9] 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.

2. Generalized Matrix Algebras

Let and be two unital algebras with unit elements and , respectively, be a faithful -bimodule, and be a -bimodule. Then,is a -algebra under matrix-like addition and multiplication, satisfying the following commutative diagrams: where and are two bimodule homomorphisms. Such a -algebra is called a generalized matrix algebra. For convenience in reading, we give the multiplication of generalized matrix algebra as follows:for all , and . Furthermore, when , then is a triangular algebra. The most common examples of generalized matrix algebras are triangular algebras and full matrix algebras (see [15, 16] for details).

Considering a generalized matrix algebra , let 1 be the unit element of . Setand . It is clear that if , then can be represented as

Hence, for any , can be represented as , where , and we can easily check that (i.e., ).

3. Main Results

In this paper, our main result is the following theorem.

Theorem 1. Let be a 2-torsion free generalized matrix algebra and be a nonlinear Jordan derivable mapping of by Lie product square-zero elements. If satisfies , then there exist an additive derivation and an additive antiderivation of , such that for all . Moreover, and are uniquely determined.

In order to prove Theorem 1, we introduce some lemmas. Next, we assume that is a 2-torsion generalized matrix algebra, is a faithful -bimodule and is a -bimodule (i.e., is a faithful -bimodule and is a -bimodule), (i.e., ), , and is a nonlinear Jordan derivable mapping of by Lie product square-zero elements.

Lemma 1. For any , then(i)(ii), , and (iii) and (iv) and

Proof. (i)Since is a nonlinear Jordan derivable mapping of by Lie product square-zero elements, then for any with , we have Therefore, taking in equation (8), we have .(ii)Since and taking in equation (8), we have Multiplying equation (9) from both the sides by and then by the property of 2-torsion freeness of , we have Multiplying equation (9) from the left by and from the right by , we get Next, we show that (). Indeed, for any , since and taking in equation (8), we obtain Multiplying equation (12) from the left by and from the right by and then by equation (10), we have ; similarly, we can get that ; therefore, we obtain from the faithfulness of that Thus, we obtain from equations (10)–(12) that () and , and then, it follows from that for all .(iii)For any (), since and taking in equation (8), it follows from Lemma 1 (ii) that Multiplying (14) by and from both sides, respectively, we get that This implies that (), and then by , we get .(iv)For any (), since () and taking in equation (8), we getMultiplying equation (16) by from both sides and then by the property of 2-torsion freeness of , we have .
Multiplying equation (16) from the left by and from the right by , it follows from Lemma 1 (ii) thatSimilarly, we can show that holds. The proof is completed.

Lemma 2. For any , then(i)(ii)(iii)(iv)(v)(vi)(vii)

Proof. (i)For any , since , taking in equation (8), and so by Lemma 1 (ii)–(iv), we have This yields that Multiplying equation (19) by from both sides and then by the property of 2-torsion freeness of and Lemma 1 (iii), we have Multiplying equation (19) from the left by and from the right by , we get Multiplying equation (19) from the left by and from the right by , we get Next, we show that . Indeed, for any , since , on the one hand, taking in equation (8), and then by Lemma 1 (iii), we get On the other hand, taking in equation (8), we get that Comparing equations (23) and (24), we get Multiplying equation (25) from the left by and from the right by , then by equation (20) and Lemma 1 (iv), we get Similarly, for any , we can get that This yields from the faithfulness of that Therefore, it follows from equations (20)–(28) and Lemma 1 (iii)-(iv) that .(ii) Similar to (i), we can show that (ii) holds.(iii)For any , since , taking in equation (8), and so by Lemma 2 (i)-(ii) and Lemma 1 (ii)–(iv), we have(iv)For any , it follows from that ; taking in equation (8) and we get from , Lemma 2 (i)-(ii), and Lemma 1 (ii)–(iv) that(v)For any , it follows from Lemma 1 (iv) that In the following, we show that . Indeed, for any , on the one hand, since and taking in equation (8), then we get On the other hand, since and , we get, respectively, Therefore, it follows from Lemma 2 (iii) and equations (34)–(36) that Multiplying equation (37) from the left by and from the right by and then by Lemma 1 (iv), we get Therefore, by the faithfulness of , we get that Hence, we get from equations (31)–(33) and (39) and Lemma 1 (iv) that . Similarly, we can show that holds for all .(vi)For any , it follows from that ; taking in equation (8) and then by Lemma 1 (ii)–(iv) and Lemma 2 (iv), we obtain This implies that In the following, we show that . Indeed, for any , on the one hand, it follows from that ; taking in equation (8) and so by and Lemma 1 (iii), we have On the other hand, since , we get Comparing equations (44) and (45), we get Multiplying equation (46) from the left by and from the right by and then by Lemma 1 (iv), equation (43), and the faithfulness of , we get Similarly, for any , we can get that Hence, for any , we get that Therefore, it follows from equations (41)–(48) and Lemma 1 (iii)-(iv) that .(vii)For any , it follows from that ; taking in equation (8) and so by Lemma 2 (vi) and Lemma 1 (iv), we haveSimilarly, we obtainTherefore, it follows from (50) and (51), Lemma 1 (ii), and the property of 2-torsion freeness of that . The proof is completed.

Lemma 3. is an additive mapping.

Proof. For any , let and , where , and by Lemma 2, we haveTherefore, is an additive mapping on .The proof is completed.

Remark 1. For any , we define a mapping asThen, by the definition of , we can easily obtain thatfor all ().
Next, we will show that is an additive antiderivation on . First, we introduce Lemma 4 and get that is an additive mapping, and then, we introduce Lemmas 5 and 6 and show that is an additive antiderivation on .

Lemma 4. Let be as in Remark 1. Then, is an additive mapping.

Proof. For any , since we have shown that is an additive mapping in Lemma 3 and then by the definition of , we obtain thatTherefore, is an additive mapping on .The proof is completed.

Lemma 5. Let be as in Remark 1. Then, for any ,(i)(ii)(iii)(iv)(v)(vi)(vii)(viii)

Proof. (i)For any , it follows from that .(ii)Similar to (i), we can showholds.(iii)For any , on the one hand, it follows from that . On the other hand, by the definition of , we have and and so ; therefore, we have .(iv)For any , on the one hand, we have . On the other hand, by the definition of , we have and , and then, it follows from that and and so .(v) Similar to (iv), we can show that (v) holds.(vi) Similar to (iv), we can show that (vi) holds.(vii)For any , since , we haveHence, by the definition of , we haveTherefore, we obtain from that . Similarly, we can show holds. The proof is completed.

Lemma 6. Let be as in Remark 3.1. Then, is an additive antiderivation.

Proof. For any , let and , where , and we obtain from Lemmas 4 and 5 thatTherefore, is an additive antiderivation on . The proof is completed.

Remark 2. For any , we define a mapping asThen, we obtain from Lemmas 3 and 4 that is an additive mapping on .
In the following, we will introduce Lemmas 7–9 and show that is an additive derivation on .

Lemma 7. Let be as in Remark 2. Then, for any ,(i) and (ii)(iii) and

Proof. By Lemma 1 and Remarks 1 and 2, we can easily check that Lemma 7 holds. The proof is completed.

Lemma 8. Let be as in Remark 2. Then, for any ,(i)(ii)(iii)(iv)(v)(vi)(vii)(viii)

Proof. (i)For any , on the one hand, since , we get that And then, it follows from , Lemma 1, and Remark 1 that On the other hand, we obtain from Lemma 6 and that Therefore, by Remark 2, equations (61) and (62), , and , we get(ii)Similarly, we can show that (ii) holds.(iii)For any , by Lemma 8 (i), on the one hand, we get On the other hand, we have Comparing equations (64) and (65), we get This yields from the faithfulness of that Furthermore, by Lemma 7 (i) and (iii), we have Therefore, we obtain from Lemma 7 (iii) and equations (67) and (68) that Similarly, we can show holds for all .(iv)For any , by Lemma 1 (iv), we have Furthermore, since , we get And so, this yields from equation (70) that Therefore, it follows from equation (72) and that(v)For any , it follows from and Lemma 7 (ii) that And, Thus, we obtain from (74) and (75) that (vi) Similar to (v), we can show that (vi) holds.(vii)For any , on the one hand, we get On the other hand, it follows from and Lemma 7 (ii)-(iii) that Comparing equations (77) and (78), we get(viii)Similar to (vii), we can show that (viii) holds. The proof is completed.

Lemma 9. Let be as in Remark 2. Then, is an additive derivation.

Proof. For any , let and , where ; since is an additive mapping and then by Lemma 8, we haveTherefore, is an additive derivation on . The proof is completed.

Proof. of Theorem 1. For any , let , where (); by Remarks 1 and 2 and Lemmas 6 and 9, we obtain thatwhere is an additive derivation and is an additive antiderivation, respectively. Furthermore, for all ().
In the following, we check that and are unique. Let be an additive derivation and be an additive antiderivation such that for all (). Suppose thatThen, for any , by Theorem 1, we get that and so for all ; this yields from and two additive derivations that is an additive derivation; moreover, by Remark 1, we get for all ().
Next, we show that for all (). Indeed, for any (), on the one hand, since is an additive derivation and () and so we get thatOn the other hand, since and are two additive antiderivations and (), we get thatThus, this yields from equations (83) and (84) that for all (). Therefore, and so and . The proof is completed.

Remark 3. By the above lemmas, Remarks 1 and 2, and the proof of Theorem 1, we can easily obtain that if is a nonlinear Jordan derivable mapping of by Lie product square-zero elements, then the following statements are equivalent:(i) is an additive derivation(ii)(iii), 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.

Corollary 1. Let and be unital algebras, be a unital -bimodule, which is faithful as both a left -module and a right -module, be a 2-torsion free triangular algebra, and be a nonlinear Jordan derivable mapping of by Lie product square-zero elements; then, is an additive derivation.

Proof. of Corollary 1. Let and be the identities of the algebras and , respectively, and let 1 be the identity of the triangular algebra . We denoteIt is clear that the triangular algebra may be represented asFor any , let , where (); then for any , since and by Lemma 1 (iii), we getTherefore, we obtain from Remark 3 (iii) that is an additive derivation. The proof is completed.

Data Availability

Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally and significantly in this paper. All authors read and approved the final manuscript. This work was supported by the National Natural Science Foundation of China (Nos. 11901451 and 11901248).

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Copyright © 2021 Xiuhai Fei and Haifang Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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