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数学进展 - 2018, Vol. 47(2): 182-188
研究论文
交换局部环上的强二和${3 \times 3}$ 矩阵
Strong 2-sum ${3\times 3}$ Matrices over Commutative Local Rings

唐高华1,2,吴严生1,3,苏华东1
TANG Gaohua1,2, WU Yansheng1,3,*, SU Huadong1

1. 广西师范学院数学与统计科学学院, 南宁, 广西, 530023;
2. 钦州学院, 钦州, 广西, 535011;
3. 南京航空航天大学理学院, 南京, 江苏, 210016
1. School of Mathematics and Statistics, Guangxi Teachers Education University, Nanning, Guangxi, 530023, P. R. China;
2. Qinzhou University, Qinzhou, Guangxi, 535011, P. R. China;
3. School of Science, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu, 210016, P. R. China

收稿日期: 2016-05-25
出版日期: 2018-05-16
2018, Vol. 47(2): 182-188
DOI: 10.11845/sxjz.2016068b

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摘要 称环 $R$ 的元有强二和性质, 如果它可以写成环中两个可交换单位的和. 如果环 $R$ 的每个元都有强二和性质, 则称环 $R$ 为强二和环. 本文研究了 $3\times3$ 阶矩阵环的两个子环$\mathcal{L}(R)$ 和 $\mathfrak{L}(R)$的强二和性质. 证明了对一交换局部环 $R$,$\mathcal{L}(R)$是强二和环当且仅当 $R$是强二和环当且仅当 $\mathfrak{L}(R)$是强二和环. 同时还证明了对一交换局部环$R$, 它是强二和环当且仅当 $\mathbb{T}_n(R)$ ($n=2,3$) 的每个角环都是强二和环.
关键词 强二和环$3\times3$矩阵环局部环角环    
Abstract:An element of a ring $R$ is called to have the strong 2-sum property if it is a sum of two units that commute with each other. And a ring $R$ is called a strong 2-sum ring if every element of $R$ has the strong 2-sum property. In this paper, we investigate the strong 2-sum property of two subrings, $ \mathcal{L}(R)$ and $\mathfrak{L}(R)$, of $3 \times 3$ matrix rings over commutative local rings. We prove that for a commutative local ring $R$, $\mathcal{L}(R)$ is a strong 2-sum ring if and only if $R$ is a strong 2-sum ring if and only if $\mathfrak{L}(R)$ is a strong 2-sum ring. Moreover, we prove that for a commutative local ring $R$, it is a strong 2-sum ring if and only if every corner ring of $\mathbb{T}_n(R) (n=2,3)$ is a strong $2$-sum ring.
Key wordsstrong 2-sum ring    $3\times 3$ matrix ring    local ring    corner ring
PACS:  O153.3  
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