# Article

Full entry | PDF   (0.2 MB)
Keywords:
commutativity preserving map; automorphism; commutative ring
Summary:
Let $\mathcal {N}=N_n(R)$ be the algebra of all $n\times n$ strictly upper triangular matrices over a unital commutative ring $R$. A map $\varphi$ on $\mathcal {N}$ is called preserving commutativity in both directions if $xy=yx\Leftrightarrow \varphi (x)\varphi (y)=\varphi (y)\varphi (x)$. In this paper, we prove that each invertible linear map on $\mathcal {N}$ preserving commutativity in both directions is exactly a quasi-automorphism of $\mathcal {N}$, and a quasi-automorphism of $\mathcal {N}$ can be decomposed into the product of several standard maps, which extains the main result of Y. Cao, Z. Chen and C. Huang (2002) from fields to rings.
References:
[1] Brešar, M.: Commuting traces of biadditive mappings, commutativity-preserving mappings and Lie mappings. Trans. Am. Math. Soc. 335 525-546 (1993). DOI 10.1090/S0002-9947-1993-1069746-X | MR 1069746 | Zbl 0791.16028
[2] Cao, Y., Chen, Z., Huang, C.: Commutativity preserving linear maps and Lie automorphisms of strictly triangular matrix space. Linear Algebra Appl. 350 41-66 (2002). MR 1906746 | Zbl 1007.15007
[3] Cao, Y., Tan, Z.: Automorphisms of the Lie algebra of strictly upper triangular matrices over a commutative ring. Linear Algebra Appl. 360 105-122 (2003). MR 1948476 | Zbl 1015.17017
[4] Marcoux, L. W., Sourour, A. R.: Commutativity preserving linear maps and Lie automorphisms of triangular matrix algebras. Linear Algebra Appl. 288 89-104 (1999). MR 1670535 | Zbl 0933.15029
[5] Omladič, M.: On operators preserving commutativity. J. Funct. Anal. 66 105-122 (1986). DOI 10.1016/0022-1236(86)90084-4 | MR 0829380 | Zbl 0587.47051
[6] Šemrl, P.: Non-linear commutativity preserving maps. Acta Sci. Math. 71 781-819 (2005). MR 2206609
[7] Wang, D., Chen, Z.: Invertible linear maps on simple Lie algebras preserving commutativity. Proc. Am. Math. Soc. 139 3881-3893 (2011). DOI 10.1090/S0002-9939-2011-10834-7 | MR 2823034 | Zbl 1258.17014
[8] Watkins, W.: Linear maps that preserve commuting pairs of matrices. Linear Algebra Appl. 14 29-35 (1976). DOI 10.1016/0024-3795(76)90060-4 | MR 0480574 | Zbl 0329.15005

Partner of