Multiplicative Skew Lie-Type Derivations on Prime ∗-Rings

Mohammad Asharf; Cihat Abdioğlu; Md Shamim Akhter; Mohammad Afajal Ansari; Mohd Shuaib Akhtar

doi:10.1007/978-981-96-9180-7_16

Multiplicative Skew Lie-Type Derivations on Prime ∗-Rings

Mohammad Asharf
, Cihat Abdioğlu
, Md Shamim Akhter
, Mohammad Afajal Ansari
, Mohd Shuaib Akhtar^*

^*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

Let R be a {2,n-1}-torsion free unital prime ∗-ring containing a nontrivial symmetric idempotent. For any x₁,x₂,…,x_n∈R, define q₁(x₁)=x₁,q₂(x₁,x₂)=[x1,x2]_∗=x₁x₂-x₂x₁^∗ and q_n(x₁,x₂,…,x_n)=[qn-1(x1,x2,…,xn-1),xn]_∗ for all integers n≥2. In this paper, we establish the relationship between multiplicative skew Lie-type derivations and additive ∗-derivations on prime ∗-rings, that is, we prove that a map δ:R→R satisfies δ(q_n(x₁,x₂,…,x_n))=∑i=1nq_n(x₁,x₂,…,x_i-1,δ(x_i),x_i+1,…,x_n)for all x₁,x₂,…,x_n∈R if and only if it is an additive ∗-derivation. As an application, multiplicative skew Lie-type derivations on factor von Neumann algebras have been characterized.

Original language	English
Title of host publication	Algebra and Differential Equations with Applications, SICMA 2023
Editors	Mohammad Ashraf, Jehad Al Jaraden, Lahcen Oukhtite, El Hassan El Kinani
Publisher	Springer
Pages	189-205
Number of pages	17
ISBN (Print)	9789819691791
DOIs	https://doi.org/10.1007/978-981-96-9180-7_16
State	Published - 2025
Externally published	Yes
Event	2nd International Conference on Mathematics and its Applications, SICMA 2023 - Antalya, Turkey Duration: 9 Nov 2023 → 13 Nov 2023

Publication series

Name	Springer Proceedings in Mathematics and Statistics
Volume	508 PROMS
ISSN (Print)	2194-1009
ISSN (Electronic)	2194-1017

Conference

Conference	2nd International Conference on Mathematics and its Applications, SICMA 2023
Country/Territory	Turkey
City	Antalya
Period	9/11/23 → 13/11/23

Bibliographical note

Publisher Copyright:
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2025.

Keywords

16W10
16W25
46L10
Additive ∗-derivation
Multiplicative skew Lie derivation
Multiplicative skew Lie-type derivation
Prime ∗-ring
von Neumann algebra

ASJC Scopus subject areas

General Mathematics

Access to Document

10.1007/978-981-96-9180-7_16

Cite this

Asharf, M., Abdioğlu, C., Akhter, M. S., Ansari, M. A., & Akhtar, M. S. (2025). Multiplicative Skew Lie-Type Derivations on Prime ∗-Rings. In M. Ashraf, J. Al Jaraden, L. Oukhtite, & E. H. El Kinani (Eds.), Algebra and Differential Equations with Applications, SICMA 2023 (pp. 189-205). (Springer Proceedings in Mathematics and Statistics; Vol. 508 PROMS). Springer. https://doi.org/10.1007/978-981-96-9180-7_16

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title = "Multiplicative Skew Lie-Type Derivations on Prime ∗-Rings",

abstract = "Let R be a \{2,n-1\}-torsion free unital prime ∗-ring containing a nontrivial symmetric idempotent. For any x1,x2,…,xn∈R, define q1(x1)=x1,q2(x1,x2)=[x1,x2]∗=x1x2-x2x1∗ and qn(x1,x2,…,xn)=[qn-1(x1,x2,…,xn-1),xn]∗ for all integers n≥2. In this paper, we establish the relationship between multiplicative skew Lie-type derivations and additive ∗-derivations on prime ∗-rings, that is, we prove that a map δ:R→R satisfies δ(qn(x1,x2,…,xn))=∑i=1nqn(x1,x2,…,xi-1,δ(xi),xi+1,…,xn)for all x1,x2,…,xn∈R if and only if it is an additive ∗-derivation. As an application, multiplicative skew Lie-type derivations on factor von Neumann algebras have been characterized.",

keywords = "16W10, 16W25, 46L10, Additive ∗-derivation, Multiplicative skew Lie derivation, Multiplicative skew Lie-type derivation, Prime ∗-ring, von Neumann algebra",

author = "Mohammad Asharf and Cihat Abdioğlu and Akhter, \{Md Shamim\} and Ansari, \{Mohammad Afajal\} and Akhtar, \{Mohd Shuaib\}",

note = "Publisher Copyright: {\textcopyright} The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2025.; 2nd International Conference on Mathematics and its Applications, SICMA 2023 ; Conference date: 09-11-2023 Through 13-11-2023",

year = "2025",

doi = "10.1007/978-981-96-9180-7\_16",

language = "English",

isbn = "9789819691791",

series = "Springer Proceedings in Mathematics and Statistics",

publisher = "Springer",

pages = "189--205",

editor = "Mohammad Ashraf and \{Al Jaraden\}, Jehad and Lahcen Oukhtite and \{El Kinani\}, \{El Hassan\}",

booktitle = "Algebra and Differential Equations with Applications, SICMA 2023",

address = "Germany",

}

Asharf, M, Abdioğlu, C, Akhter, MS, Ansari, MA & Akhtar, MS 2025, Multiplicative Skew Lie-Type Derivations on Prime ∗-Rings. in M Ashraf, J Al Jaraden, L Oukhtite & EH El Kinani (eds), Algebra and Differential Equations with Applications, SICMA 2023. Springer Proceedings in Mathematics and Statistics, vol. 508 PROMS, Springer, pp. 189-205, 2nd International Conference on Mathematics and its Applications, SICMA 2023, Antalya, Turkey, 9/11/23. https://doi.org/10.1007/978-981-96-9180-7_16

Multiplicative Skew Lie-Type Derivations on Prime ∗-Rings. / Asharf, Mohammad; Abdioğlu, Cihat; Akhter, Md Shamim et al.
Algebra and Differential Equations with Applications, SICMA 2023. ed. / Mohammad Ashraf; Jehad Al Jaraden; Lahcen Oukhtite; El Hassan El Kinani. Springer, 2025. p. 189-205 (Springer Proceedings in Mathematics and Statistics; Vol. 508 PROMS).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

TY - GEN

T1 - Multiplicative Skew Lie-Type Derivations on Prime ∗-Rings

AU - Asharf, Mohammad

AU - Abdioğlu, Cihat

AU - Akhter, Md Shamim

AU - Ansari, Mohammad Afajal

AU - Akhtar, Mohd Shuaib

N1 - Publisher Copyright: © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2025.

PY - 2025

Y1 - 2025

N2 - Let R be a {2,n-1}-torsion free unital prime ∗-ring containing a nontrivial symmetric idempotent. For any x1,x2,…,xn∈R, define q1(x1)=x1,q2(x1,x2)=[x1,x2]∗=x1x2-x2x1∗ and qn(x1,x2,…,xn)=[qn-1(x1,x2,…,xn-1),xn]∗ for all integers n≥2. In this paper, we establish the relationship between multiplicative skew Lie-type derivations and additive ∗-derivations on prime ∗-rings, that is, we prove that a map δ:R→R satisfies δ(qn(x1,x2,…,xn))=∑i=1nqn(x1,x2,…,xi-1,δ(xi),xi+1,…,xn)for all x1,x2,…,xn∈R if and only if it is an additive ∗-derivation. As an application, multiplicative skew Lie-type derivations on factor von Neumann algebras have been characterized.

AB - Let R be a {2,n-1}-torsion free unital prime ∗-ring containing a nontrivial symmetric idempotent. For any x1,x2,…,xn∈R, define q1(x1)=x1,q2(x1,x2)=[x1,x2]∗=x1x2-x2x1∗ and qn(x1,x2,…,xn)=[qn-1(x1,x2,…,xn-1),xn]∗ for all integers n≥2. In this paper, we establish the relationship between multiplicative skew Lie-type derivations and additive ∗-derivations on prime ∗-rings, that is, we prove that a map δ:R→R satisfies δ(qn(x1,x2,…,xn))=∑i=1nqn(x1,x2,…,xi-1,δ(xi),xi+1,…,xn)for all x1,x2,…,xn∈R if and only if it is an additive ∗-derivation. As an application, multiplicative skew Lie-type derivations on factor von Neumann algebras have been characterized.

KW - 16W10

KW - 16W25

KW - 46L10

KW - Additive ∗-derivation

KW - Multiplicative skew Lie derivation

KW - Multiplicative skew Lie-type derivation

KW - Prime ∗-ring

KW - von Neumann algebra

UR - https://www.scopus.com/pages/publications/105028249128

U2 - 10.1007/978-981-96-9180-7_16

DO - 10.1007/978-981-96-9180-7_16

M3 - Conference contribution

AN - SCOPUS:105028249128

SN - 9789819691791

T3 - Springer Proceedings in Mathematics and Statistics

SP - 189

EP - 205

BT - Algebra and Differential Equations with Applications, SICMA 2023

A2 - Ashraf, Mohammad

A2 - Al Jaraden, Jehad

A2 - Oukhtite, Lahcen

A2 - El Kinani, El Hassan

PB - Springer

T2 - 2nd International Conference on Mathematics and its Applications, SICMA 2023

Y2 - 9 November 2023 through 13 November 2023

ER -

Multiplicative Skew Lie-Type Derivations on Prime ∗-Rings

Abstract

Publication series

Conference

Bibliographical note

Keywords

ASJC Scopus subject areas

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