Characterization of Lie-type higher derivations of triangular rings

Mohammad Ashraf; Mohammad Afajal Ansari; Md Shamim Akhter

doi:10.1515/gmj-2022-2195

Published by De Gruyter October 26, 2022

Characterization of Lie-type higher derivations of triangular rings

Mohammad Ashraf , Mohammad Afajal Ansari and Md Shamim Akhter

From the journal Georgian Mathematical Journal

https://doi.org/10.1515/gmj-2022-2195

Showing a limited preview of this publication:

Abstract

Let 𝔄 be a triangular ring and let p n ⁢ ( U 1 , U 2 , … , U n ) denote the ( n - 1 ) th commutator of elements U 1 , U 2 , … , U n ∈ 𝔄 . Suppose that ℕ is the set of nonnegative integers and 𝔏 = { ξ r } r ∈ ℕ is a sequence of additive mappings on 𝔄 such that ξ 0 = i ⁢ d 𝔄 , the identity mapping on 𝔄 , and for each r ∈ ℕ ,

ξ r ⁢ ( p n ⁢ ( U 1 , U 2 , … , U n ) ) = ∑ i 1 + i 2 + ⋯ + i n = r p n ⁢ ( ξ i 1 ⁢ ( U 1 ) , ξ i 2 ⁢ ( U 2 ) , … , ξ i n ⁢ ( U n ) )

for all U 1 , U 2 , … , U n ∈ 𝔄 with U 1 ⁢ U 2 ⁢ ⋯ ⁢ U n = 0 . In this paper, it is shown that under certain conditions 𝔏 = { ξ r } r ∈ ℕ has the standard form, that is, there exist a higher derivation { d r } r ∈ ℕ on 𝔄 and a family { h r } r ∈ ℕ of additive mappings h r : 𝔄 → 𝒵 ⁢ ( 𝔄 ) satisfying h r ⁢ ( p n ⁢ ( U 1 , U 2 , … , U n ) ) = 0 for all U 1 , U 2 , … , U n ∈ 𝔄 with U 1 ⁢ U 2 ⁢ ⋯ ⁢ U n = 0 such that for each r ∈ ℕ , ξ r ⁢ ( U ) = d r ⁢ ( U ) + h r ⁢ ( U ) for all U ∈ 𝔄 .

Keywords: Derivation; Lie derivation; Lie higher derivation; Lie-type higher derivation; triangular ring,nest algebra

MSC 2010: 16W25; 47L35; 15A78

Funding statement: This research is partially supported by a research grant from NBHM (No. 02011/5/2020NBHM (R.P.) R & D II/6243) and a research grant from DST (No. DST/INSPIRE/03/2017/IF170834).

Acknowledgements

The authors are indebted to the referee for his/her valuable comments and suggestions.

References

[1] I. Z. Abdullaev, n-Lie derivations on von Neumann algebras (in Russian), Uzbek. Mat. Zh. (1992), no. 5–6, 3–9. Search in Google Scholar

[2] J. Alaminos, J. Extremera, A. R. Villena and M. Brešar, Characterizing homomorphisms and derivations on C * -algebras, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007), no. 1, 1–7. 10.1017/S0308210505000090Search in Google Scholar

[3] M. A. Ansari, M. Ashraf and M. S. Akhtar, Lie triple derivations on trivial extension algebras, Bull. Iranian Math. Soc. 48 (2022), no. 4, 1763–1774. 10.1007/s41980-021-00618-3Search in Google Scholar

[4] M. Ashraf, M. S. Akhtar and M. A. Ansari, Generalized Lie (Jordan) triple derivations on arbitrary triangular algebras, Bull. Malays. Math. Sci. Soc. 44 (2021), no. 6, 3767–3776. 10.1007/s40840-021-01148-1Search in Google Scholar

[5] M. Ashraf, M. S. Akhtar and A. Jabeen, Characterizations of Lie triple higher derivations of triangular algebras by local actions, Kyungpook Math. J. 60 (2020), no. 4, 683–710. Search in Google Scholar

[6] M. Ashraf and A. Jabeen, Characterization of Lie type derivation on von Neumann algebra with local actions, Bull. Korean Math. Soc. 58 (2021), no. 5, 1193–1208. Search in Google Scholar

[7] D. Benkovič and D. Eremita, Multiplicative Lie n-derivations of triangular rings, Linear Algebra Appl. 436 (2012), no. 11, 4223–4240. 10.1016/j.laa.2012.01.022Search in Google Scholar

[8] M. Brešar, Characterizing homomorphisms, derivations and multipliers in rings with idempotents, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007), no. 2, 9–21. 10.1017/S0308210504001088Search in Google Scholar

[9] S. U. Chase, A generalization of the ring of triangular matrices, Nagoya Math. J. 18 (1961), 13–25. 10.1017/S0027763000002208Search in Google Scholar

[10] M. A. Chebotar, W.-F. Ke and P.-H. Lee, Maps characterized by action on zero products, Pacific J. Math. 216 (2004), no. 2, 217–228. 10.2140/pjm.2004.216.217Search in Google Scholar

[11] W.-S. Cheung, Commuting maps of triangular algebras, J. Lond. Math. Soc. (2) 63 (2001), no. 1, 117–127. 10.1112/S0024610700001642Search in Google Scholar

[12] W.-S. Cheung, Lie derivations of triangular algebras, Linear Multilinear Algebra 51 (2003), no. 3, 299–310. 10.1080/0308108031000096993Search in Google Scholar

[13] Y. Ding and J. Li, Lie n-higher derivations and Lie n-higher derivable mappings, Bull. Aust. Math. Soc. 96 (2017), no. 2, 223–232. 10.1017/S0004972717000338Search in Google Scholar

[14] Y. Du and Y. Wang, Lie derivations of generalized matrix algebras, Linear Algebra Appl. 437 (2012), no. 11, 2719–2726. 10.1016/j.laa.2012.06.013Search in Google Scholar

[15] D. Han, Lie-type higher derivations on operator algebras, Bull. Iranian Math. Soc. 40 (2014), no. 5, 1169–1194. Search in Google Scholar

[16] D. Han and F. Wei, Multiplicative Lie higher derivations of unital algebras with idempotents, Oper. Matrices 10 (2016), no. 2, 345–377. 10.7153/oam-10-19Search in Google Scholar

[17] J. Hou and X. Qi, Additive maps derivable at some points on J -subspace lattice algebras, Linear Algebra Appl. 429 (2008), no. 8–9, 1851–1863. 10.1016/j.laa.2008.05.013Search in Google Scholar

[18] P. Ji and W. Qi, Characterizations of Lie derivations of triangular algebras, Linear Algebra Appl. 435 (2011), no. 5, 1137–1146. 10.1016/j.laa.2011.02.048Search in Google Scholar

[19] P. Ji, W. Qi and X. Sun, Characterizations of Lie derivations of factor von Neumann algebras, Linear Multilinear Algebra 61 (2013), no. 3, 417–428. 10.1080/03081087.2012.689982Search in Google Scholar

[20] W. Jing, On Jordan all-derivable points of B ⁢ ( H ) , Linear Algebra Appl. 430 (2009), no. 4, 941–946. 10.1016/j.laa.2008.09.006Search in Google Scholar

[21] W. Jing, S. Lu and P. Li, Characterisations of derivations on some operator algebras, Bull. Aust. Math. Soc. 66 (2002), no. 2, 227–232. 10.1017/S0004972700040077Search in Google Scholar

[22] J. Li and Q. Shen, Characterizations of Lie higher and Lie triple derivations on triangular algebras, J. Korean Math. Soc. 49 (2012), no. 2, 419–433. 10.4134/JKMS.2012.49.2.419Search in Google Scholar

[23] L. Liu, Lie triple derivations on factor von Neumann algebras, Bull. Korean Math. Soc. 52 (2015), no. 2, 581–591. 10.4134/BKMS.2015.52.2.581Search in Google Scholar

[24] F. Lu and W. Jing, Characterizations of Lie derivations of B ⁢ ( X ) , Linear Algebra Appl. 432 (2010), no. 1, 89–99. 10.1016/j.laa.2009.07.026Search in Google Scholar

[25] X. Qi, Characterizing Lie n-derivations for reflexive algebras, Linear Multilinear Algebra 63 (2015), no. 8, 1693–1706. 10.1080/03081087.2014.968519Search in Google Scholar

[26] X. Qi and J. Hou, Characterization of Lie derivations on prime rings, Comm. Algebra 39 (2011), no. 10, 3824–3835. 10.1080/00927872.2010.512588Search in Google Scholar

[27] X. Qi and J. Hou, Characterization of Lie derivations on von Neumann algebras, Linear Algebra Appl. 438 (2013), no. 1, 533–548. 10.1016/j.laa.2012.08.019Search in Google Scholar

[28] X. Qi and J. Ji, Characterizing derivations on von Neumann algebras by local actions, J. Funct. Spaces Appl. 2013 (2013), Article ID 407427. 10.1155/2013/407427Search in Google Scholar

[29] X. F. Qi, Characterization of Lie higher derivations on triangular algebras, Acta Math. Sin. (Engl. Ser.) 29 (2013), no. 5, 1007–1018. 10.1007/s10114-012-1548-3Search in Google Scholar

[30] X.-F. Qi and J.-C. Hou, Lie higher derivations on nest algebras, Commun. Math. Res. 26 (2010), no. 2, 131–143. Search in Google Scholar

[31] Y. Wang, Lie n-derivations of unital algebras with idempotents, Linear Algebra Appl. 458 (2014), 512–525. 10.1016/j.laa.2014.06.029Search in Google Scholar

Received: 2022-02-03

Accepted: 2022-04-05

Published Online: 2022-10-26

Published in Print: 2023-02-01

Characterization of Lie-type higher derivations of triangular rings

Abstract

Acknowledgements

References

Journal and Issue

Articles in the same Issue