Jordan delta-Derivations of Associative Algebras | OMICS International
ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
Make the best use of Scientific Research and information from our 700+ peer reviewed, Open Access Journals that operates with the help of 50,000+ Editorial Board Members and esteemed reviewers and 1000+ Scientific associations in Medical, Clinical, Pharmaceutical, Engineering, Technology and Management Fields.
Meet Inspiring Speakers and Experts at our 3000+ Global Conferenceseries Events with over 600+ Conferences, 1200+ Symposiums and 1200+ Workshops on
Medical, Pharma, Engineering, Science, Technology and Business

# Jordan delta-Derivations of Associative Algebras

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

#### Abstract

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.

#### Keywords

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

#### Introduction

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

#### Jordan δ-derivations of Associative Algebras

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 Ra, where Ra -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 ] = me13 for some m ∈ B. So, for a = t(e12 + e21) 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.

#### Jordan δ-pre-derivations of Associative Algebras

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.

#### Jordan δ-derivations of Triangular Algebras

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 = 1B, 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.

#### References

Select your language of interest to view the total content in your interested language

### Recommended Conferences

• 7th International Conference on Biostatistics and Bioinformatics
September 26-27, 2018 Chicago, USA
• Conference on Biostatistics and Informatics
December 05-06-2018 Dubai, UAE
• Mathematics Congress - From Applied to Derivatives
December 5-6, 2018 Dubai, UAE

### Article Usage

• Total views: 12004
• [From(publication date):
specialissue-2015 - Jul 17, 2018]
• Breakdown by view type
• HTML page views : 8076

## Post your comment

Peer Reviewed Journals

Make the best use of Scientific Research and information from our 700 + peer reviewed, Open Access Journals
International Conferences 2018-19

Meet Inspiring Speakers and Experts at our 3000+ Global Annual Meetings

### Conferences By Subject

Agri & Aquaculture Journals

Dr. Krish

+1-702-714-7001Extn: 9040

Biochemistry Journals

Datta A

1-702-714-7001Extn: 9037

Business & Management Journals

Ronald

1-702-714-7001Extn: 9042

Chemistry Journals

Gabriel Shaw

1-702-714-7001Extn: 9040

Clinical Journals

Datta A

1-702-714-7001Extn: 9037

Engineering Journals

James Franklin

1-702-714-7001Extn: 9042

Food & Nutrition Journals

Katie Wilson

1-702-714-7001Extn: 9042

General Science

Andrea Jason

1-702-714-7001Extn: 9043

Genetics & Molecular Biology Journals

Anna Melissa

1-702-714-7001Extn: 9006

Immunology & Microbiology Journals

David Gorantl

1-702-714-7001Extn: 9014

Materials Science Journals

Rachle Green

1-702-714-7001Extn: 9039

Nursing & Health Care Journals

Stephanie Skinner

1-702-714-7001Extn: 9039

Medical Journals

Nimmi Anna

1-702-714-7001Extn: 9038

Neuroscience & Psychology Journals

Nathan T

1-702-714-7001Extn: 9041

Pharmaceutical Sciences Journals

Ann Jose

1-702-714-7001Extn: 9007

Social & Political Science Journals

Steve Harry

1-702-714-7001Extn: 9042

© 2008- 2018 OMICS International - Open Access Publisher. Best viewed in Mozilla Firefox | Google Chrome | Above IE 7.0 version