Abstract

In this paper, we proved that each nonlinear nonglobal semi-Jordan triple derivable mapping on a 2-torsion free triangular algebra is an additive derivation. As its application, we get the similar conclusion on a nest algebra or a 2-torsion free block upper triangular matrix algebra, respectively.

1. Introduction

Let be a commutative ring with identity and a unital algebra over , , and be an additive mapping on . For any , we denote the Jordan product of by . For any , if implies , then is said a 2-torsion free algebra. Recall that is called a derivation if for all ; is called a Jordan derivation if , for all ; is called a triple derivation if , for all . is called a Jordan triple derivation if , for all . Furthermore, if is without assumption of additivity in the above definitions, then is said a nonlinear (triple) derivable mapping and a nonlinear Jordan (triple) derivable mapping, respectively. Obviously, every derivation is a Jordan derivation, every derivation is a triple derivation, and every triple derivation is a Jordan triple derivation. However, the inverse statement is not true in general.

A natural and very interesting problem that we are dealing with is studying certain conditions on an algebra such that each Jordan (triple) derivation (nonlinear Jordan (triple) derivable mapping) is a derivation.

In the past few decades, many mathematicians studied this problem and obtained abundant results. For example, Herstein, in [1], proved that every Jordan derivation on a prime ring not of characteristic 2 is a derivation. This result was extended by Cusack in [2] and Brešar and Vukman in [3] to the case of semiprime, respectively. Zhang, in [4, 5], showed that every Jordan derivation on a nest algebra or a 2-torsion free triangular algebra is an inner derivation or a derivation, respectively. Later, Ghahramani, in [6], extended the result of Zhang and Yu in [5] and proved that, under certain conditions, each Jordan derivation on trivial extension algebras is a sum of a derivation and an antiderivation. For other similar results about Jordan derivations (nonlinear Jordan derivable mappings), we refer the readers to [7–9] and references therein, for more details. With the deepening of research, many research achievements have been obtained about Jordan triple derivations and nonlinear Jordan triple derivable mappings. For example, Bresar, in [10], proved that every Jordan triple derivation on a 2-torsion-free semiprime ring is a derivation. Similar conclusion have been obtained in [11] by Bell and Kappe. Zhao and Li, in [12], proved that every nonlinear -Jordan triple derivation on von Neumann algebras is an additive -derivation. For other similar results about Jordan triple derivations (nonlinear Jordan triple derivable mappings), we refer the readers to [13–15] and references therein, for more details.

In 2016, Ashraf and Jabeen, in [15], obtained that if is without the additivity assumption and satisfiesfor all , then such a is an additive derivation on a 2-torsion-free triangular algebra.

In this paper, we call that is a nonlinear nonglobal semi-Jordan triple derivable mapping on if is without the additivity assumption and satisfiesfor all with .

Here, it needs to be pointed out that our above definition is different from Ashraf’s and Jabeen’s in [15]. We will discuss the nonlinear nonglobal semi-Jordan triple derivable mappings on triangular algebras and obtain one main result (see Theorem 1).

For convenient reading, we give some basic concepts and properties of triangular algebras as follows.

Let and be unital algebras over a commutative ring and be a unital -bimodule, which is faithful as both are a left -module and a right -module. Then, the -algebra,under the usual matrix operations is called a triangular algebra. We refer the reader to [16] for more details about the triangular algebras. Basic examples of triangular algebras are upper triangular matrix algebras and nest algebras.

Let and be the identities of the algebras and , respectively, and let 1 be the identity of the triangular algebra . Throughout this paper, we shall use the following notations:

It is clear that the triangular algebra may be represented aswhere and are subalgebras of isomorphic to and , respectively, and is a -bimodule isomorphic to the -bimodule .

2. Nonlinear Nonglobal Semi-Jordan Triple Derivable Mappings on Triangular Algebras

In this section, our main result is Theorem 1, and we will show Theorem 1 holds.

Theorem 1. Let be a 2-torsion-free triangular algebra and be a mapping from into itself (without assumption of additivity) such thatfor all with . Then, is an additive derivation.
In order to prove Theorem 1, we introduce Lemmas 1–5 and then prove that Lemmas 1–5 hold. We assume that be a 2-torsion-free triangular algebra , and be a nonlinear nonglobal semi-Jordan triple derivable mapping on triangular algebra .

Lemma 1. and .

Proof. Taking in equation (6), we have . Since , taking in equation (6), we obtainThis yields from the property of 2-torsion freeness of thatSimilarly, we obtain thatFor any , since , taking in equation (6), we haveMultiplying the above equation from left by and from right by , it follows from that . Similarly, we obtain that . Therefore, according to the faithfulness of and the property of 2-torsion free of , we haveSo, by equations (8)–(11), we have . The proof is completed.

Lemma 2. For any , then
(i) and
(ii) and
(iii)

Proof. (i) For any , since , taking in equation (6), then it follows from thatThis implies that . Furthermore, multiplying the above equation from left by and from right by , it follows from thatSimilarly, we can show (ii) holds.
(iii) For any , since , taking equation (6), we get from thatTherefore, . The proof is completed.

Lemma 3. For any , , and , then
(i)
(ii)
(iii)
(iv)
(v)

Proof. (i) For any and , since , taking in equation (6), it follows from Lemma 2 that(ii) For any and , since , taking in equation (6), we can obtain from and Lemma 2 thatSimilarly, we can show (iii) holds.
(iv) For any , by Lemma 2 (ii), on the one hand, we get thatOn the other hand,Comparing the above two equations, we obtainThis yields from the faithfulness of and Lemma 2 thatFurthermore, by Lemma 2 (i), we obtain thatTherefore, by the above two equations and , we get . Similarly, we can show () holds. The proof is completed.

Lemma 4. For any , , and , then
(i)
(ii)
(iii)
(iv)
(v)

Proof. (i) For any , , and , since , taking in equation (6), then by and , we obtainThen, it follows from and the property of 2-torsion freeness of thatFurthermore, since , taking in equation (6), then we obtain from Lemma 2, , and thatThis yields from Lemma 3 (ii) that , and then, by the faithfulness of , we get thatHence, by equations (23)–(31) and Lemma 2, we get . Similarly, we can show (ii) holds.
(iii) For any , since , taking in equation (6), then by Lemmas 2 and 4 (i)-(ii) and , we get that(iv) For any , since , taking in equation (6), then it follows from thatTherefore, this implies from Lemmas 4 (iii) and 3 (ii) that , so by the faithfulness of , we get thatFurthermore, for any , we can get from Lemma 2 (i) thatTherefore, we get from above two equations and that . Similarly, we can show () holds. The proof is completed.