The group of commutativity preserving maps on strictly upper triangular matrices
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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 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.
Keywordscommutativity preserving map automorphism commutative ring
MSC 201017C30 15A04 15A27 15A99 13C10
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