Medical, Pharma, Engineering, Science, Technology and Business

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.

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

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

Alternative ring; Idempotent element; Maps; Additivity

In this paper, 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, z* ∈, to reduce the number of parentheses. For *x; y; z* ∈ we denote the associator by (*x*, *y*, *z*)=(*xy*)*z*-*x*(*yz*).

A ring is called k-torsion free if *kx*=0 implies *x*=0; for any *x* ∈ ; where, prime if *IJ*≠0 for any two nonzero ideals *I, J* ⊆ and semiprime if it contains no nonzero ideal whose square is zero.

A ring is said to be alternative if,

(*x, x, y*)=0=(*y, x, x*), for all* x, y* ∈

and flexible if,

(*x, y, x*)=0, for all *x, y* ∈

One easily sees that any alternative ring is flexible.

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

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

A mapping is Jordan triple multiplicative derivation if,

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

for all *a*, *b* ∈ . Let us consider an alternative ring and let us *x* a nontrivial idempotent is not an unity element. Let be given by *e*_{2}*a*=*a* – *e*_{1}*a* and . We shall denote by *ae*_{2}. Note that need not have an identity element. The operation *x*(1 – *y*) for *x*, *y* ∈ is understood as *x* – *xy*: It is easy to see that and *i*, *j*=1, 2. Then has a Peirce decomposition , where (*i*, *j*=1, 2), satisfying the multiplicative relations:

**Remark 1.1**

By the linearization of (iv) we obtain,

*x _{ij}y_{ij}*+

if *i* ≠ *j*. 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 be an alternative ring containing a non-trivial idempotent *e*_{1} and , the Peirce Decomposition of ; relative to *e*_{1}, satisfying:

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

And,

**Theorem 1.3**

Let be an alternative ring containing a non-trivial idempotent *e*_{1} and , the Peirce Decomposition of , relative to *e*_{1}, satisfying:

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

We shall prove the following result:

**Theorem 2.1**

Let be an alternative ring containing a non-trivial idempotent *e*_{1} and , the Peirce Decomposition of ; relative to *e*_{1}, satisfying:

for * i*, *j* 2 {1; 2}. If 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:**

**Proof:**

**Lemma 2.2:**

**Proof:** For any *x _{ij}* ∈ ,

On the other hand,

These imply that,

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

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

By Condition (iii), we see that,

Equivalently,

**Lemma 2.3:**

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

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

To Prove (2) just use the identity.

**Lemma 2.4: **

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

we can get,

This implies, by conditions of the Theorem 2.1 that,

This completes the proof.

**Lemma 2.5:**

**Proof:** Let a_{kk} and b_{kk} be arbitrary elements of , *k*=1; 2. By considering for the cases of *i* ≠*j* and *i*=*j* respectively, one can easily get that,

where in the second identity *i*=*j* ≠*k*: Now we have only to prove that,

with *k*=*j*. For any , from,

*r*_{11}*x*_{12}=(*e*_{1}+*r*_{11})*x*_{12}(*e*_{1}+*r*_{11}) (1)

and,

*x*_{12}*r*_{22}=(*e*_{1}+*r*_{22})*x*_{12}(*e*_{1}+*r*_{22}) (2)

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

(3)

(4)

Now, applying equality (3) for *r*_{11}=*a*_{11}+*b*_{11}, *r*_{11}=*a*_{11} and *r*_{11}=*b*_{11} and applying equality (4) for *r*_{22}=*a*_{22}+*b*_{22}, *r*_{22}=*a*_{22} and *r*_{22}=*b*_{22}, we can get,

It follows from Condition (i) and (ii) of the Theorem 2.1 that , 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*; *b* ∈ , we write *a*=*a*_{11}+*a*_{12}+*a*_{21}+*a*_{22} and *b*=*b*_{11}+*b*_{12}+*b*_{21}+*b*_{22}. Applying the previous Lemmas, we have,

In the case of an unital alternative ring we have.

**Corollary 3.1**

Let be an unital alternative ring containing a non-trivial idempotent *e*_{1} and ; the Peirce Decomposition of ; relative to *e*_{1}; satisfying:

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

As a last result of our paper follows the Corollary.

**Corollary 3.2**

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

for all *a*, *b* ∈ , then *D* is additive.

- Slater M (1970) Prime Alternative Rings-I. Journal of Algebra 15: 229-243.
- Ferreira BLM, Ferreira JCM (2016) Additivity of n-Multiplicative Maps on Alternative Rings. Communications In Algebra 44: 1557-1568.
- Ferreira BLM, Ferreira JCM, Guzzo H (2013) Jordan Maps on Alternative Algebras. JP Journal of Algebra, Number Theory and Applications 31: 129-142.
- Ferreira BLM, Ferreira JCM, Guzzo H (2014) Jordan Triple Elementary Maps on Alternative Rings. Extracta Mathematicae 29: 1-18.
- Ferreira BLM, Ferreira JCM, Guzzo H (2014) Jordan Triple Maps of Alternative Algebras. JP Journal of Algebra, Number Theory and Application 33: 25-33.
- Ferreira BLM, Nascimento R (2014) Derivable Maps on Alternative Rings. Recen16: 9-15.
- Ferreira RN, Ferreira BLM (2017) Jordan Derivation on Alternative Rings. International Journal of Mathematics, Game Theory and Algebra 25: 435-445.

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