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arXiv 1107.0384math.RA
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关于具有被加数和性质的环的一个注记

A note on rings with the summand sum property

Liang Shen

备注：
7 pages
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摘要：

如果$R_{R}$的任意两个直被加数的和（交集）也是直被加数，则环$R$被称为右SSP（SIP）。左侧可以类似地定义。以下是等价的：（1）$R$是右SSP。(2)$R$是正确的C3和正确的SIP。(3)$R$位于左侧C3和左侧SIP。(4)$R$留在SSP。（2）$R$是半单环当且仅当列有限矩阵环$\mathbb{C}\mathbb{F}\mathbb{M}_Λ(R)$对于可数无限集$Λ$是右SSP当且仅当列有限矩阵环$\mathbb{C}\mathbb{F}\mathbb{M}_Λ(R)$对于任意无限集$Λ$是右SSP.改进了已有的一些结果。

英文摘要：

A ring $R$ is called right SSP (SIP) if the sum (intersection) of any two direct summands of $R_{R}$ is also a direct summand. Left sides can be defined similarly. The following are equivalent: (1) $R$ is right SSP. (2) $R$ is right C3 and right SIP. (3) $R$ is left C3 and left SIP. (4) $R$ is left SSP. It is also shown that (1) $R$ is a von-Neumann regular ring if and only if $\mathbb{M}_{2}(R)$ is right SSP if and only if $\mathbb{M}_{n}(R)$ is right SSP for some $n>1$; (2) $R$ is a semisimple ring if and only if the column finite matrix ring $\mathbb{C}\mathbb{F}\mathbb{M}_Λ(R)$ is right SSP for a countably infinite set $Λ$ if and only if the column finite matrix ring $\mathbb{C}\mathbb{F}\mathbb{M}_Λ(R)$ is right SSP for any infinite set $Λ$. Some known results are improved.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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