The group of commutativity preserving maps on strictly upper triangular matrices

Deng Yin Wang; Min Zhu; Jianling Rou

Czechoslovak Mathematical Journal (2014)

  • Volume: 64, Issue: 2, page 335-350
  • ISSN: 0011-4642

Abstract

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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 x y = y x ⇔ 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.

How to cite

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Wang, Deng Yin, Zhu, Min, and Rou, Jianling. "The group of commutativity preserving maps on strictly upper triangular matrices." Czechoslovak Mathematical Journal 64.2 (2014): 335-350. <http://eudml.org/doc/262031>.

@article{Wang2014,
abstract = {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.},
author = {Wang, Deng Yin, Zhu, Min, Rou, Jianling},
journal = {Czechoslovak Mathematical Journal},
keywords = {commutativity preserving map; automorphism; commutative ring; commutativity preserving map; automorphism; commutative ring},
language = {eng},
number = {2},
pages = {335-350},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The group of commutativity preserving maps on strictly upper triangular matrices},
url = {http://eudml.org/doc/262031},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Wang, Deng Yin
AU - Zhu, Min
AU - Rou, Jianling
TI - The group of commutativity preserving maps on strictly upper triangular matrices
JO - Czechoslovak Mathematical Journal
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 2
SP - 335
EP - 350
AB - 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.
LA - eng
KW - commutativity preserving map; automorphism; commutative ring; commutativity preserving map; automorphism; commutative ring
UR - http://eudml.org/doc/262031
ER -

References

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  1. Brešar, M., 10.1090/S0002-9947-1993-1069746-X, Trans. Am. Math. Soc. 335 525-546 (1993). (1993) Zbl0791.16028MR1069746DOI10.1090/S0002-9947-1993-1069746-X
  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). (2002) Zbl1007.15007MR1906746
  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). (2003) Zbl1015.17017MR1948476
  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). (1999) Zbl0933.15029MR1670535
  5. Omladič, M., 10.1016/0022-1236(86)90084-4, J. Funct. Anal. 66 105-122 (1986). (1986) Zbl0587.47051MR0829380DOI10.1016/0022-1236(86)90084-4
  6. Šemrl, P., Non-linear commutativity preserving maps, Acta Sci. Math. 71 781-819 (2005). (2005) MR2206609
  7. Wang, D., Chen, Z., 10.1090/S0002-9939-2011-10834-7, Proc. Am. Math. Soc. 139 3881-3893 (2011). (2011) Zbl1258.17014MR2823034DOI10.1090/S0002-9939-2011-10834-7
  8. Watkins, W., 10.1016/0024-3795(76)90060-4, Linear Algebra Appl. 14 29-35 (1976). (1976) Zbl0329.15005MR0480574DOI10.1016/0024-3795(76)90060-4

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