Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

STRONG COMMUTATIVITY PRESERVING MAPS OF UPPER TRIANGULAR MATRIX LIE ALGEBRAS OVER A COMMUTATIVE RING

  • Chen, Zhengxin (School of Mathematics and Statistic Fujian Normal University) ;
  • Zhao, Yu'e (School of Mathematics and Statistic Qingdao University)
  • Received : 2020.09.08
  • Accepted : 2021.03.31
  • Published : 2021.07.31
Download PDF
⟨ Previous Next ⟩

Abstract

Let R be a commutative ring with identity 1, n ≥ 3, and let 𝒯n(R) be the linear Lie algebra of all upper triangular n × n matrices over R. A linear map 𝜑 on 𝒯n(R) is called to be strong commutativity preserving if [𝜑(x), 𝜑(y)] = [x, y] for any x, y ∈ 𝒯n(R). We show that an invertible linear map 𝜑 preserves strong commutativity on 𝒯n(R) if and only if it is a composition of an idempotent scalar multiplication, an extremal inner automorphism and a linear map induced by a linear function on 𝒯n(R).

Keywords

Acknowledgement

This work is supported by the National Natural Science Foundation of China (Grant No. 11871014) and the Natural Science Foundation of Fujian Province (Grant No. 2020J01162).

References

  1. H. E. Bell and M. N. Daif, On commutativity and strong commutativity-preserving maps, Canad. Math. Bull. 37 (1994), no. 4, 443-447. https://doi.org/10.4153/CMB-1994-064-x
  2. D. Benkovic and D. Eremita, Commuting traces and commutativity preserving maps on triangular algebras, J. Algebra 280 (2004), no. 2, 797-824. https://doi.org/10.1016/j.jalgebra.2004.06.019
  3. M. Bresar, Commuting traces of biadditive mappings, commutativity-preserving mappings and Lie mappings, Trans. Amer. Math. Soc. 335 (1993), no. 2, 525-546. https://doi.org/10.2307/2154392
  4. M. Bresar and C. R. Miers, Strong commutativity preserving maps of semiprime rings, Canad. Math. Bull. 37 (1994), no. 4, 457-460. https://doi.org/10.4153/CMB-1994-066-4
  5. M. Bresar and P. Semrl, Commutativity preserving linear maps on central simple algebras, J. Algebra 284 (2005), no. 1, 102-110. https://doi.org/10.1016/j.jalgebra.2004.09.014
  6. Y. Cao, Z. Chen, and C. Huang, Commutativity preserving linear maps and Lie automorphisms of strictly triangular matrix space, Linear Algebra Appl. 350 (2002), 41-66. https://doi.org/10.1016/S0024-3795(02)00264-1
  7. Z. Chen and H. Liu, Strong commutativity preserving maps of strictly triangular matrix Lie algebras, J. Algebra Appl. 18 (2019), no. 7, 1950134, 15 pp. https://doi.org/10.1142/S0219498819501342
  8. M. D. Choi, A. A. Jafarian, and H. Radjavi, Linear maps preserving commutativity, Linear Algebra Appl. 87 (1987), 227-241. https://doi.org/10.1016/0024-3795(87)90169-8
  9. Q. Deng and M. Ashraf, On strong commutativity preserving mappings, Results Math. 30 (1996), no. 3-4, 259-263. https://doi.org/10.1007/BF03322194
  10. V. De Filippis and G. Scudo, Strong commutativity and Engel condition preserving maps in prime and semiprime rings, Linear Multilinear Algebra 61 (2013), no. 7, 917-938. https://doi.org/10.1080/03081087.2012.716433
  11. C.-K. Li and N.-K. Tsing, Linear preserver problems: a brief introduction and some special techniques, Linear Algebra Appl. 162/164 (1992), 217-235. https://doi.org/10.1016/0024-3795(92)90377-M
  12. J.-S. Lin and C.-K. Liu, Strong commutativity preserving maps on Lie ideals, Linear Algebra Appl. 428 (2008), no. 7, 1601-1609. https://doi.org/10.1016/j.laa.2007.10.006
  13. J.-S. Lin and C.-K. Liu, Strong commutativity preserving maps in prime rings with involution, Linear Algebra Appl. 432 (2010), no. 1, 14-23. https://doi.org/10.1016/j.laa.2009.06.036
  14. C.-K. Liu, Strong commutativity preserving generalized derivations on right ideals, Monatsh. Math. 166 (2012), no. 3-4, 453-465. https://doi.org/10.1007/s00605-010-0281-1
  15. J. Ma, X. W. Xu, and F. W. Niu, Strong commutativity-preserving generalized derivations on semiprime rings, Acta Math. Sin. (Engl. Ser.) 24 (2008), no. 11, 1835-1842. https://doi.org/10.1007/s10114-008-7445-0
  16. L. W. Marcoux and A. R. Sourour, Commutativity preserving linear maps and Lie automorphisms of triangular matrix algebras, Linear Algebra Appl. 288 (1999), no. 1-3, 89-104. https://doi.org/10.1016/S0024-3795(98)10182-9
  17. D. Wang, Invertible linear maps on Borel subalgebras preserving zero Lie products, J. Algebra 406 (2014), 321-336. https://doi.org/10.1016/j.jalgebra.2014.02.023
  18. D. Wang and Z. Chen, Invertible linear maps on simple Lie algebras preserving commutativity, Proc. Amer. Math. Soc. 139 (2011), no. 11, 3881-3893. https://doi.org/10.1090/S0002-9939-2011-10834-7