Jordan Triple Derivation on Alternative Rings

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.


Introduction
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 k ∈ , k>0, 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 R be a 3-torsion free alternative ring. So R is a prime ring if and only if aR⋅b=0 (or a⋅R b=0) implies a=0 or b=0 for a, b ∈ R.
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 e 1 ∈ , i.e, 2

Remark 1.1
By the linearization of (iv) we obtain, 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][3][4][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 = 11 ⊕ 12 ⊕ 21  22 , the Peirce Decomposition of ; relative to e 1 , satisfying: And,

Theorem 1.3
Let  be an alternative ring containing a non-trivial idempotent e 1 and = 11 ⊕ 12 ⊕ 21  22 , the Peirce Decomposition of , relative to e 1 , satisfying: for i, j, k ∈{1, 2}. If D:  → is a Jordan multiplicative derivation, then D is additive.
The proof of the Theorem is organized as a series of Lemmas.

Applications in Prime Alternative Rings
In the case of an unital alternative ring we have.
As a last result of our paper follows the Corollary.
for all a, b ∈ , then  is additive.