用对易子描述四元数

Mathematical Proceedings of the Royal Irish Academy Pub Date : 2021-07-21 DOI:10.1353/mpr.2022.0000

E. Kleinfeld, Yoav Segev

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引用次数: 1

摘要

摘要:设R是一个不可交换的带1的结合环。设任意非零对易子v∈R满足:v2在R的中心，且v不是零因子。证明了R不存在零因子，且如果char(R)≠2，则R在其中心的定域是一个四元数除法代数。我们的证明是初等的和自成体系的。

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A Characterisation of the Quaternions Using Commutators

Abstract:

Let R be an associative ring with 1, which is not commutative. Assume that any non-zero commutator v ∈ R satisfies: v2 is in the centre of R, and v is not a zero divisor.

We prove that R has no zero divisors, and that if char(R) ≠ 2, then the localisation of R at its centre is a quaternion division algebra. Our proof is elementary and self contained.

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Mathematical Proceedings of the Royal Irish Academy

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