Geodesic
The group of commutativity preserving maps on strictly upper triangular matrices
Czechoslovak Mathematical Journal, Tome 64 (2014) no. 2, pp. 335-350 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
DOI : 10.1007/s10587-014-0105-x
Classification : 13C10, 15A04, 15A27, 15A99, 17C30
Keywords: commutativity preserving map; automorphism; commutative ring 
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     title = {The group of commutativity preserving maps on strictly upper triangular matrices},
     journal = {Czechoslovak Mathematical Journal},
     pages = {335--350},
     year = {2014},
     volume = {64},
     number = {2},
     doi = {10.1007/s10587-014-0105-x},
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     zbl = {06391498},
     language = {en},
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Wang, Dengyin; Zhu, Min; Rou, Jianling. The group of commutativity preserving maps on strictly upper triangular matrices. Czechoslovak Mathematical Journal, Tome 64 (2014) no. 2, pp. 335-350. doi: 10.1007/s10587-014-0105-x

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