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ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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Jordan Triple Derivation on Alternative Rings

Ferreira BLM* and Ferreira RN

Centro de Matematica, Computacao e Cognicao, Universidade Federal do ABC, Av. dos Estados 5001, 09210-580, Santo Andre, Brazil

*Corresponding Author:
Ferreira BLM
Centro de Matematica, Computacao e Cognicao
Universidade Federal do ABC, Av. dos Estados 5001
09210-580, Santo Andre, Brazil
Tel: +55 (41) 3310-4545
E-mail: [email protected]

Received Date: May 26, 2017; Accepted Date: July 26, 2017; Published Date: July 31, 2017

Citation: Ferreira BLM, Ferreira RN (2017) Jordan Triple Derivation on Alternative Rings. J Generalized Lie Theory Appl 11: 275. doi: 10.4172/1736-4337.1000275

Copyright: © 2017 Ferreira BLM, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

Let D be a mapping from an alternative ring Equation into itself satisfying D(a⋅ba)= D(a)⋅ba+aD(b)a+abD(a) for all a, b Equation. Under some conditions on Equation, we show that D is additive.

Keywords

Alternative ring; Idempotent element; Maps; Additivity

Introduction

In this paper, Equation will be a ring not necessarily associative or commutative and consider the following convention for its multiplication operation: xy⋅z=(xy)z and x⋅yz=x(yz) for x, y, zEquation, to reduce the number of parentheses. For x; y; zEquation we denote the associator by (x, y, z)=(xy)z-x(yz).

A ring Equation is called k-torsion free if kx=0 implies x=0; for any xEquation ; whereEquation, prime if IJ≠0 for any two nonzero ideals I, JEquation and semiprime if it contains no nonzero ideal whose square is zero.

A ring Equation is said to be alternative if,

(x, x, y)=0=(y, x, x), for all x, yEquation

and flexible if,

(x, y, x)=0, for all x, yEquation

One easily sees that any alternative ring is flexible.

Theorem 1.1 Let Equation be a 3-torsion free alternative ring. So Equation is a prime ring if and only if Equation implies a=0 or b=0 for Equation

Proof: Clearly all alternative rings satisfying the properties Equation are prime rings. Suppose Equation is a prime ring by [1] Lemma 2:4, Theorem A and Proposition 3:5] we have Equation is a chain of subrings of Equation. If Equation hence Equation follows [1] Proposition 3.5 (e)] that a=0 or b=0.

A mapping Equation is Jordan triple multiplicative derivation if,

Equation

for all a, bEquation. It is worth noting that by the flexible identity of alternative rings we can write,

Equation

for all a, bEquation. Let us consider Equation an alternative ring and let us x a nontrivial idempotent Equation is not an unity element. Let Equation be given by e2a=ae1a and Equation . We shall denote Equation by ae2. Note that Equation need not have an identity element. The operation x(1 – y) for x, yEquation is understood as xxy: It is easy to see that Equation and i, j=1, 2. Then Equation has a Peirce decomposition Equation, where Equation (i, j=1, 2), satisfying the multiplicative relations:

Equation

Equation

Remark 1.1

By the linearization of (iv) we obtain,

xijyij+yijxij=0

if ij. This identity is very useful for the main result to be verified.

The study of the relationship between the multiplicative and the additive structures of a ring has become an interesting and active topic in ring theory. In non-associative ring theory we can mention recent works such as [2-5] where the authors generalized the results for a class of non-associative rings, namely alternative rings. The present paper we investigate the problem of when a Jordan triple multiplicative derivation must be an additive map for the class of alternative rings. The hypotheses of the main Theorem allow the author to make its proof based on calculus using the Peirce decomposition notion for Alternative rings. But it is worth noting that the notion of Peirce decomposition for the alternative rings is similar to the notion of Peirce decomposition for the associative rings. However, the similarity of this notion is only in its written form, but not in its theoretical structure because the Peirce decomposition for alternative rings is the generalization of the Peirce decomposition for associative rings. The symbol “⋅”, as defined in the introduction section of our article, is essential to elucidate how the non-associative multiplication should be done, and also the symbol “⋅” is used to simplify the notation. Therefore, the symbol “⋅” is crucial to the logic, characterization and generalization of associative results to the alternative results. In this paper we shall continue the line of research introduced in refs. [6,7] where its authors demonstrate the following results.

Theorem 1.2

Let Equation be an alternative ring containing a non-trivial idempotent e1 and Equation, the Peirce Decomposition of Equation; relative to e1, satisfying:

Equation

for i, j, k ∈ {1, 2}. If Equation is a multiplicative derivation, then D is additive.

And,

Theorem 1.3

Let Equation be an alternative ring containing a non-trivial idempotent e1 and Equation, the Peirce Decomposition of Equation, relative to e1, satisfying:

Equation

for i, j, k ∈ {1, 2}. If Equation is a Jordan multiplicative derivation, then D is additive.

The Main Theorem

We shall prove the following result:

Theorem 2.1

Let Equation be an alternative ring containing a non-trivial idempotent e1 and Equation, the Peirce Decomposition of Equation; relative to e1, satisfying:

Equation

for i, j 2 {1; 2}. If Equation is a Jordan triple multiplicative derivation, then D is additive.

The proof of the Theorem is organized as a series of Lemmas.

We begin with the following Lemma with a simple proof.

Lemma 2.1: Equation

Proof: Equation

Lemma 2.2: Equation

Proof: For any xijEquation, i, j=1, 2, on one hand, we have,

Equation

On the other hand,

These imply that,

Equation

where we use the flexible identity. By the flexible identity we note that for any i, j=1, 2, we have,

Equation

Then, for i, j=1, 2, we get,

Equation

By Condition (iii), we see that,

Equation

Equivalently,

Equation

Lemma 2.3: Equation

Equation

Proof: We will prove only (1) because the proof of (2) is similar to (1). By Remark 1.1 we note that,

Equation

By applying Lemma 2.2, Remark 1.1 and flexible identity we have,

Equation

To Prove (2) just use the identity.

Equation

Lemma 2.4: Equation

Proof: For any Equation, i, j=1, 2, from,

Equation

we can get,

Equation

This implies, by conditions of the Theorem 2.1 that,

Equation

This completes the proof.

Lemma 2.5: Equation

Proof: Let akk and bkk be arbitrary elements of Equation, k=1; 2. By considering Equation for the cases of ij and i=j respectively, one can easily get that,

Equation

where in the second identity i=jk: Now we have only to prove that,

Equation

with k=j. For any Equation, from,

r11x12=(e1+r11)x12(e1+r11) (1)

and,

x12r22=(e1+r22)x12(e1+r22) (2)

can check, by (1) and (2) that,

Equation (3)

Equation (4)

Now, applying equality (3) for r11=a11+b11, r11=a11 and r11=b11 and applying equality (4) for r22=a22+b22, r22=a22 and r22=b22, we can get,

Equation

It follows from Condition (i) and (ii) of the Theorem 2.1 that Equation, with k=j, which completes the proof.a

Now we are ready to prove our main result.

Proof of the Theorem 2.1: For any a; bEquation, we write a=a11+a12+a21+a22 and b=b11+b12+b21+b22. Applying the previous Lemmas, we have,

Equation

Applications in Prime Alternative Rings

In the case of an unital alternative ring we have.

Corollary 3.1

Let Equation be an unital alternative ring containing a non-trivial idempotent e1 and Equation; the Peirce Decomposition of Equation; relative to e1; satisfying:

Equation

Equation

for i; j ∈ {1; 2}. If Equation is a Jordan triple multiplicative derivation, then D is additive.

As a last result of our paper follows the Corollary.

Corollary 3.2

Let Equation be a 3-torsion free prime unital alternative ring with a nontrivial idempotent. If mapping Equation satisfies,

Equation

for all a, bEquation, then D is additive.

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