Document Type: Research Paper

Author

Department of Mathematics‎, ‎Centre R'{e}gional des M'{e}tiers‎ ‎d'Education et de Formation (CRMEF) Tangier‎, ‎Morocco

Abstract

‎Let $\mathcal{F}$ be an field of zero characteristic and $N_{\infty‎}(‎\mathcal{F})$ be the algebra of infinite strictly upper triangular‎ ‎matrices with entries in $\mathcal{F}$‎, ‎and $f:N_{\infty}(\mathcal{F}‎)\rightarrow N_{\infty}(\mathcal{F})$ be a non-additive Lie centralizer of $‎N_{\infty }(\mathcal{F})$; that is‎, ‎a map satisfying that $f([X,Y])=[f(X),Y]$‎ ‎for all $X,Y\in N_{\infty}(\mathcal{F})$‎. ‎We prove that $f(X)=\lambda X$‎, ‎where $\lambda \in \mathcal{F}$‎.

Keywords

Main Subjects

[1] J. Bounds, Commuting maps over the ring of strictly upper triangular matrices, Linear Algebra Appl. 507 (2016), 132-136.

[2] M. Bresar, Centralizing mappings and derivations in prime rings, J. Algebra 156 (1993), 385-394.

[3] M. Bresar, Commuting traces of biadditive mappings, commutativity-preserving mappings and Lie mappings, Trans. Amer. Math. Soc. 335 (1993), 525-546.

[4] W. S. Cheung, Commuting maps of triangular algebras, J. London Math. Soc. 63 (2) (2001), 117-127.

[5] D. Eremita, Commuting traces of upper triangular matrix rings, Aequat. Math. 91 (2017), 563-578.

[6] A. Fosner, W. Jing, Lie centralizers on triangular rings and nest algebras, Adv. Oper. Theory, in press.

[7] W. Franca, Commuting maps on some subsets of matrices that are not closed under addition, Linear Algebra Appl. 437 (2012), 388-391.

[8] F. Ghomanjani, M. A. Bahmani, A note on Lie centralizer maps, Palest. J. Math. 7 (2) (2018), 468-471.

[9] R. Slowik, Expressing infinite matrices as products of involutions, Linear Algebra Appl. 438 (2013), 399-404.

[10] J. Vukman, An identity related to centralizers in semiprime rings, Comment. Math. Univ. Carolin. 40 (3) (1999), 447-456.

[11] B. Zalar, On centralizers of semiprime rings, Comment. Math. Univ. Carolin. 32 (4) (1991), 609-614.