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| 摘要: |
| 本文研究了每个元素均可表示为三个互相交换的三幂等元之和的约化环, 给出了整环上任意\ $n$ 阶矩阵均可表示为三个三幂等矩阵之和的判定条件, 证明了对于整环\ $R$, $M_{n}(R)$ 中任意矩阵可分解为三个三幂等矩阵之和当且仅当\ $R\cong \mathbb{Z}_{p}$, 其中\ $p=2, 3, 5$ 或\ $7$. 文中所得结果推进了文献[Abyzov A N, Tapkin D T. When is every matrix over a ring the sum of two tripotents? Linear Algebra Appl., 2021]中的主要结果 |
| 关键词: 三幂等元 友矩阵 约化环 矩阵环 |
| DOI： |
| 分类号:O153.3 |
| 基金项目:Key Laboratory of Financial Mathematics of Fujian Province University; NSF of AnhuiProvince |
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| MATRICES OVER A REDUCED RING AS SUMS OF THREE TRIPOTENTS |
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huangtao,cuijian
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| Abstract: |
| In this paper, we study reduced rings in which every element is a sum of three tripotents that commute, and determine the integral domains over
which every $n\times n$ matrix is a sum of three tripotents.
It is proved that for an integral domain $R$, every matrix in $M_{n}(R)$ is a sum of three tripotents if
and only if $R\cong \mathbb{Z}_{p}$ with $p=2, 3, 5$ or $7$. Our theorems somewhat
considerably improved on results in [Abyzov A N, Tapkin D T. When is every matrix over a ring the sum of two tripotents? Linear Algebra Appl., 2021]. |
| Key words: Tripotent companion matrix reduced ring matrix ring |
