Abstract

In this paper, we describe the structure of triple derivations of trivial extensions. We then apply our results to triangular rings.

1. Introduction

Let be a ring with identity, and be the center of . Let be a unitary -bimodule. An additive mapping from into is said to be a derivation if for all . The additive mapping is called a biderivation if it is a derivation in each component; that is, and are fulfilled for all . Likewise, one can further develop the definition of triple derivations in an analogous manner. An additive mapping is called a triple derivation if it is a derivation in each component; that is,for all . The trivial extension of by is the ringwith the componentwise addition and the multiplication given by

Trivial extensions have been extensively studied in algebra and analysis (see, for instance, [17]).

Let and be rings with identity, be a unitary -bimodule, and be the triangular ring determined by , and with the usual addition and multiplication of matrices (see [8, 9] for more details of triangular rings). Then, one can easily verify that can be made into a unitary -bimodule via the scalar multiplications given by

Hence, is the trivial extension of by . It is clear that triangular rings are examples of trivial extensions. In [1], derivations and biderivations of trivial extensions are studied. Besides the afore-mentioned work, biderivations in different backgrounds are studied in [1012], where further references can be found. A natural question arises: What is the representation of -derivations on trivial extensions? Since the study of -derivations is too complex, we study 3-derivations on trivial extensions and conjecture the representation of -derivations.

2. Main Results and Proofs

We first describe triple derivations of the trivial extension .

Theorem 1. Let be a triple derivation of a trivial extension , . Then, there exist(i)triple derivations of and (ii)mappings , , , , , (iii)mappings , , , , , (iv)a mapping which is a bimodule homomorphism in each coordinate(v)a mapping Such thatholds for all and .

Proof. Let and . For any , we haveTherefore, and are derivations in the first coordinate. Similarly, we get that and are derivations in the second and third coordinates. This proves (i).
To prove (ii) and (iii), let and set . Let be also in . Then,It follows from the above expression that are derivations in the first coordinate. Similarly, defineWe get that are derivations in the coordinate of the element in , . Moreover, we havefor all . On the other hand, we getfor all . According to the above relations, we obtain that is a bimodule homomorphism in the second coordinate, andfor all . Likewise, is a bimodule homomorphism in the coordinate of the element in andfor all .
To prove (iv) and (v), let be arbitrary, and assumeSincefor all , we have that is a bimodule homomorphism in the first coordinate, andfor all . Similarly, we get that is a bimodule homomorphism in the second and third coordinates, andfor all .
In conclusion, we getfor all and .
According to Theorem 1, we can decompose the triple derivation on into the sum of five triple derivations.

Corollary 1. Let and be as above. Then, the mappings defined byare triple derivations of , and .
Next, we will use Theorem 1 to study triple derivations of a triangular ring identified with the trivial extension .

Theorem 2. Let be a triple derivation of the trivial extension . Then,(i)there exist triple derivations of and of such that, for every and , we have(ii)there exists an element such that, for every and , we have(iii)the mappings , ,(iv)for every and , we havewhere are mappings(v)for every , we have

Proof. (i) Since is a triple derivation on , we havefor all . It follows thatSimilarly, we get thatSet . In view of (40), we haveThis implies that . On the other hand, we haveThis yields that . Hence, . Likewise,Assume that . By (41), we obtainTherefore, we define . It follows thatThus, is a derivation in the first coordinate. Similarly, is a derivation in the second and third coordinates, so that is a triple derivation of . By an analogue computation, one can show that there exists a triple derivation of such thatfor all . Hence, .
(ii) Assume that , we haveSince is a triple derivation, it follows thatfor all . According to the above relation, we havefor all . This yields thatfor all . Similarly, we get thatfor all . Analogue computation shows that, for every and , we haveThis proves (ii).
(iii) Since is a bimodule homomorphism in each coordinate, we haveSince is a bimodule homomorphism in the coordinate of the element in , we get , .
(iv) Recalling that is a derivation in the first coordinate, for every , we haveSet . It follows that . Using (55) and (56), we haveSimilarly, there exist two mappings such thatfor all .
Set . Since is a derivation in the first and second coordinates, it follows that