Non-global Multiplicative Lie Triple Derivations on Rings

Mohammad Ashraf; Mohammad Afajal Ansari; Md Shamim Akhter

doi:10.1007/978-3-031-50795-3_15

Non-global Multiplicative Lie Triple Derivations on Rings

Mohammad Ashraf
, Mohammad Afajal Ansari^*
, Md Shamim Akhter

^*Corresponding author for this work

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

Abstract

Let R be a ring containing a nontrivial idempotent with center Z(R). In the present article, it is shown that under certain restrictions every map ξ:R→R (not necessarily additive) satisfying ξ([[S,T],U])=[[ξ(S),T],U]+[[S,ξ(T)],U]+[[S,T],ξ(U)] for all S,T,U∈R with STU=0, is almost additive, that is, ξ(S+T)-ξ(S)-ξ(T)∈Z(R). In addition, if R is a 2-torsion free prime ring, then ξ is of the form ξ=∂+η, where ∂ is a derivation from R into its central closure S and η is a map from R into its extended centroid C such that η(S+T)-η(S)-η(T)∈Z(R) and η([[S,T],U])=0 for all S,T,U∈R with STU=0. The obtained results are then applied to standard operator algebras, factor von Neumann algebras and the algebra of all bounded linear operators.

Original language	English
Title of host publication	Advances in Ring Theory and Applications - WARA22
Editors	Shakir Ali, Mohammad Ashraf, Nadeem ur Rehman, Vincenzo De Filippis
Publisher	Springer
Pages	207-222
Number of pages	16
ISBN (Print)	9783031507946
DOIs	https://doi.org/10.1007/978-3-031-50795-3_15
State	Published - 2024
Externally published	Yes
Event	Workshop on Associative Rings and Algebras with Additional Structures, WARA 2022 - Messina, Italy Duration: 18 Jul 2022 → 20 Jul 2022

Publication series

Name	Springer Proceedings in Mathematics and Statistics
Volume	443
ISSN (Print)	2194-1009
ISSN (Electronic)	2194-1017

Conference

Conference	Workshop on Associative Rings and Algebras with Additional Structures, WARA 2022
Country/Territory	Italy
City	Messina
Period	18/07/22 → 20/07/22

Bibliographical note

Publisher Copyright:
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024.

Keywords

Derivation
Multiplicative Lie triple derivation
Ring
Standard operator algebra
Von Neumann algebra

ASJC Scopus subject areas

General Mathematics

Access to Document

10.1007/978-3-031-50795-3_15

Cite this

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title = "Non-global Multiplicative Lie Triple Derivations on Rings",

abstract = "Let R be a ring containing a nontrivial idempotent with center Z(R). In the present article, it is shown that under certain restrictions every map ξ:R→R (not necessarily additive) satisfying ξ([[S,T],U])=[[ξ(S),T],U]+[[S,ξ(T)],U]+[[S,T],ξ(U)] for all S,T,U∈R with STU=0, is almost additive, that is, ξ(S+T)-ξ(S)-ξ(T)∈Z(R). In addition, if R is a 2-torsion free prime ring, then ξ is of the form ξ=∂+η, where ∂ is a derivation from R into its central closure S and η is a map from R into its extended centroid C such that η(S+T)-η(S)-η(T)∈Z(R) and η([[S,T],U])=0 for all S,T,U∈R with STU=0. The obtained results are then applied to standard operator algebras, factor von Neumann algebras and the algebra of all bounded linear operators.",

keywords = "Derivation, Multiplicative Lie triple derivation, Ring, Standard operator algebra, Von Neumann algebra",

author = "Mohammad Ashraf and Ansari, \{Mohammad Afajal\} and Akhter, \{Md Shamim\}",

note = "Publisher Copyright: {\textcopyright} The Author(s), under exclusive license to Springer Nature Switzerland AG 2024.; Workshop on Associative Rings and Algebras with Additional Structures, WARA 2022 ; Conference date: 18-07-2022 Through 20-07-2022",

year = "2024",

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Ashraf, M, Ansari, MA & Akhter, MS 2024, Non-global Multiplicative Lie Triple Derivations on Rings. in S Ali, M Ashraf, NU Rehman & V De Filippis (eds), Advances in Ring Theory and Applications - WARA22. Springer Proceedings in Mathematics and Statistics, vol. 443, Springer, pp. 207-222, Workshop on Associative Rings and Algebras with Additional Structures, WARA 2022, Messina, Italy, 18/07/22. https://doi.org/10.1007/978-3-031-50795-3_15

Non-global Multiplicative Lie Triple Derivations on Rings. / Ashraf, Mohammad; Ansari, Mohammad Afajal; Akhter, Md Shamim.
Advances in Ring Theory and Applications - WARA22. ed. / Shakir Ali; Mohammad Ashraf; Nadeem ur Rehman; Vincenzo De Filippis. Springer, 2024. p. 207-222 (Springer Proceedings in Mathematics and Statistics; Vol. 443).

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

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T1 - Non-global Multiplicative Lie Triple Derivations on Rings

AU - Ashraf, Mohammad

AU - Ansari, Mohammad Afajal

AU - Akhter, Md Shamim

N1 - Publisher Copyright: © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024.

PY - 2024

Y1 - 2024

N2 - Let R be a ring containing a nontrivial idempotent with center Z(R). In the present article, it is shown that under certain restrictions every map ξ:R→R (not necessarily additive) satisfying ξ([[S,T],U])=[[ξ(S),T],U]+[[S,ξ(T)],U]+[[S,T],ξ(U)] for all S,T,U∈R with STU=0, is almost additive, that is, ξ(S+T)-ξ(S)-ξ(T)∈Z(R). In addition, if R is a 2-torsion free prime ring, then ξ is of the form ξ=∂+η, where ∂ is a derivation from R into its central closure S and η is a map from R into its extended centroid C such that η(S+T)-η(S)-η(T)∈Z(R) and η([[S,T],U])=0 for all S,T,U∈R with STU=0. The obtained results are then applied to standard operator algebras, factor von Neumann algebras and the algebra of all bounded linear operators.

AB - Let R be a ring containing a nontrivial idempotent with center Z(R). In the present article, it is shown that under certain restrictions every map ξ:R→R (not necessarily additive) satisfying ξ([[S,T],U])=[[ξ(S),T],U]+[[S,ξ(T)],U]+[[S,T],ξ(U)] for all S,T,U∈R with STU=0, is almost additive, that is, ξ(S+T)-ξ(S)-ξ(T)∈Z(R). In addition, if R is a 2-torsion free prime ring, then ξ is of the form ξ=∂+η, where ∂ is a derivation from R into its central closure S and η is a map from R into its extended centroid C such that η(S+T)-η(S)-η(T)∈Z(R) and η([[S,T],U])=0 for all S,T,U∈R with STU=0. The obtained results are then applied to standard operator algebras, factor von Neumann algebras and the algebra of all bounded linear operators.

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M3 - Conference contribution

AN - SCOPUS:85193611626

SN - 9783031507946

T3 - Springer Proceedings in Mathematics and Statistics

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EP - 222

BT - Advances in Ring Theory and Applications - WARA22

A2 - Ali, Shakir

A2 - Ashraf, Mohammad

A2 - Rehman, Nadeem ur

A2 - De Filippis, Vincenzo

PB - Springer

T2 - Workshop on Associative Rings and Algebras with Additional Structures, WARA 2022

Y2 - 18 July 2022 through 20 July 2022

ER -

Non-global Multiplicative Lie Triple Derivations on Rings

Abstract

Publication series

Conference

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

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Cite this