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**Ivan Kaygorodov ^{*}**

Institute of Mathematics, Novosibirsk, Russia

- Corresponding Author:
- Ivan Kaygorodov

Institute of Mathematics Novosibirsk, Russia

**Tel:**(383) 333-28-92

**E-mail:**[email protected]

**Received date:** July 21, 2015 **Accepted date:** August 03, 2015 **Published date:** August 31, 2015

**Citation:** Kaygorodov I (2015) Jordan δ-Derivations of Associative Algebras. J Generalized Lie Theory Appl S1:003. doi:10.4172/1736-4337.S1-003

**Copyright:** © 2015 Kaygorodov I. 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

We described the structure of jordan δ-derivations and jordan δ-prederivations of unital associative algebras. We gave examples of nonzero jordan 1/2 -derivations, but not 1/2 -derivations.

δ-derivation; Jordan δ-derivation; Associative algebra; Triangular algebras

Let Jordan δ-derivation be a generalization of the notion of jordan derivation [1,2] and **δ-derivation **[3-14]. Jordan δ-derivation is a linear mappings j, for a fixed element of δ from the main field, satisfies the following condition

Note that various generalizations of Jordan derivations have been widely studied [15-17]. If algebra A is a (anti) commutetive algebra, then jordan δ-derivation of A is a δ-derivation of A.

In this paper we consider jordan δ-derivations of associative unital algebras. Naturally, we are interested in the nonzero mappings with

and algebras over field with characteristic . In the main body of work, we using the following standard notation

In this chapter, we consider jordan δ-derivations of associative unital algebras. And prove, that **jordan δ-derivation **of simple associative unital algebra is a δ- derivation. Also, we give the example of non-trivial jordan -derivations

**Lemma:** Let A be an unital associative algebra and j be a jordan δ-derivation, then and , where where [x, [x, a]] = 0 for any x ∈ A.

**Proof:** Let x = 1 in condition (1), then j(1) = 0 or If j(1) = 0, then for x = y + 1 in (1), we get

That is, if j(1) = 0, then j(y) = 0.

If and j(1) = a, then Using the identity (1), obtain

and

That is [x, [x, a]] = 0. Lemma is proved.

It is easy to see, that mapping , where [x, [x, a]] = 0 for any x ∈ A, is a jordan -derivation. Using Kaygorodov et al. [6] -derivation of unital associative algebra A is a mapping R_{a}, where R_{a} -multiplication by the element in the center of the algebra A.

Below we give an example of an unital associative algebra with a derivation, different from derivation.

**Example:** Consider the algebra of upper triangular matrices of size 3×3 with zero diagonal over a non-commutative algebra B. Let A^{#} be an algebra with an adjoined identity for the algebra A. Then, easy to see, that for any elements X, Y ∈ A^{#}, right [X, Y ] = me_{13} for some m ∈ B. So, for a = t(e_{12} + e_{21}) and t ∈ B, will be but [X, [X, a]] = 0. So, using corollary from Lemma, mapping is a jordan - derivation of algebra A^{#}, but not - derivation of algebra A^{#} and

**Theorem 1: **Jordan δ-derivation of simple unital

A is a δ-derivation.

**Proof: **Note, that case of δ = 1 was study in Herstein et al. Cusack et al. [1,2]. It is clear, that the case is more interesting. Using Herstein et al. [18], is a simple Lie algebra. Clearly, that [[a, x], x] = 0 and [[x, a], a] = 0. Using roots system of simple Lie algebra [19], we can obtain, that a ∈ Z (A), so [A, a] = 0. Which implies that the mapping j is a derivation. Theorem is proved.

Linear mapping ζ be a prederivation of algebra A, if for any elements x, y, z ∈ A:

ζ(xyz) = (x)yz + xζ(y)z + xyζ(z).

Prederivations considered in Burde and Bajo et al. [20, 21]. Jordan δ-prederivation ς is a linear mapping, satisfies the following condition

The main purpose of this section is showing that Jordan δ-prederivation of unital **associative algebra **is a jordan derivation or jordan derivation

**Theorem 2: **Let ς be a jordan δ-prederivation of unital associative algebra A, then ς is a jordan - derivation or jordan derivation.

**Proof:** Note, that if ς is a jordan δ-prederivation, then So, ς (1) = 0 or If

That is, we have

Replace x by x + 1, then obtain

So, using (2), we obtain

That is

We easily obtain

Replace x by x + 1, then obtain [x, [x, a]] = 0. Using Lemma, we obtain that ς is a jordan -derivation. The case ς (1) = 0 is treated similarly, and the basic calculations are omitted. In this case, we obtain that ς is a jordan derivation (for δ = 1) or zero mapping. Theorem is proved.

Let A and B be unital associative algebras over a field R and M be an unital (A, B)-bimodule, which is a left A-module and right B-module. The R-algebra

under the usual matrix operations will be called a triangular algebra. This kind of algebras was first introduced by Chase [22]. Actively studied the derivations and their generalization to **triangular algebras **[15-17, 23].

Triangular algebra is an unital associative algebra and triangular algebras satisfy the conditions of Lemma. So, if j is a jordan δ-derivation

of algebra T, then and there is C, which where for any X ∈ T.

Also,

for any . Easy to see, mapping satisfing condition and satisfing condition are jordan derivations, respectively, of algebras

A and B. Also, for m = 0, y = 0 and x = 1A we can get

m_{*} = 0.

On the other hand, for x = 0 and y = 1_{B}, we can get

mb = am.

**Theorem 3: **Let A and B be a central simple algebras, then jordan δ-derivation of triangular algebra T is a δ-derivation.

Proof: T is an unital algebra and we can consider case of Algebras A and B are central simple algebras, then and Using (4), we obtain . So, jordan - derivation of T is a - derivation

Theorem is proved.

**Theorem 4: **Let A be a central simple algebra and M be a faithful module right B-module, then jordan δ-derivation of triangular T is a δ-derivation.

**Proof:** T is an unital algebra and we can consider case of Algebra A is a central simple algebra, then Using (4), we obtain αm = mb. The module M is a faithful module, we have So, jordan - derivation is a -derivation. Theorem is proved.

Comment: Noted, using the example if non-trivial jordan -derivation, but not -derivation, of unital associative algebra, we can construct new example of non-trivial jordan -derivation of triangular algebra. For example, we can consider triangular algebra ,where A^{#} is a bimodule ovar A#. In conclusion, the author expresses his gratitude to Prof. Pavel Kolesnikov for interest and constructive comments.

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