The group of commutativity preserving maps on strictly upper triangular matrices. (English) Zbl 1340.15004
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 extends the main result of Y. Cao, Z. Chen and C. Huang [Linear Algebra Appl. 350, No. 1–3, 41–66 (2002; Zbl 1007.15007)] from fields to rings.
MSC:
| 15A04 | Linear transformations, semilinear transformations |
| 15A30 | Algebraic systems of matrices |
| 15A86 | Linear preserver problems |
| 16S50 | Endomorphism rings; matrix rings |
Citations:
Zbl 1007.15007References:
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