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The group of commutativity preserving maps on strictly upper triangular matrices

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Abstract

Let N = N n (R) be the algebra of all n × n strictly upper triangular matrices over a unital commutative ring R. A map φ on N is called preserving commutativity in both directions if xy = yx ⇔ φ(x)φ(y) = φ(y)φ(x). In this paper, we prove that each invertible linear map on N preserving commutativity in both directions is exactly a quasi-automorphism of N, and a quasi-automorphism of 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.

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Authors and Affiliations

  1. Department of Mathematics, China University of Mining and Technology, Xuzhou, 221008, P.R. China

    Dengyin Wang, Min Zhu & Jianling Rou

Authors
  1. Dengyin Wang
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  2. Min Zhu
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  3. Jianling Rou
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Corresponding author

Correspondence to Dengyin Wang.

Additional information

Supported by the Fundamental Research Funds for the Central Universities (2012LWA08).

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Wang, D., Zhu, M. & Rou, J. The group of commutativity preserving maps on strictly upper triangular matrices. Czech Math J 64, 335–350 (2014). https://doi.org/10.1007/s10587-014-0105-x

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  • DOI: https://doi.org/10.1007/s10587-014-0105-x

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