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arXiv 1005.4167math.RA
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关于可数$Σ$ - C2环

On countably $Σ$-C2 rings

Liang Shen, Jianlong Chen

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9 pages
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摘要：

设$R$为环。称$R$为右可数$Σ$环，如果$R_{R}$的每个可数直和拷贝是C2模.以下对于环$R$是等价的：（1）$R$是右可数$Σ$ -C2环。(2)列有限矩阵环$\mathbb{C}\mathbb{F}\mathbb{M}_{\mathbb{N}}(R)$是右C2（或C3）环. (3)每个$R_{R}$的可数直和拷贝是C3模。(4)每个投射右$R$ -模是C2（或C3）模. (5)$R$是右完全环，且$R_{R}$的每个有限直和副本是C2（或C3）模.证明了右可数$Σ$ -C 2环就是Hyman Bass在1960年引入的右有限投射维数r $FPD(R)$ =sup\{ $Pd_{R}(M)|$ $M$是右$R$ -模且$Pd_{R}(M)<\infty$ \}=0的环.

英文摘要：

Let $R$ be a ring. $R$ is called a right countably $Σ$-C2 ring if every countable direct sum copies of $R_{R}$ is a C2 module. The following are equivalent for a ring $R$: (1) $R$ is a right countably $Σ$-C2 ring. (2) The column finite matrix ring $\mathbb{C}\mathbb{F}\mathbb{M}_{\mathbb{N}}(R)$ is a right C2 (or C3) ring. (3) Every countable direct sum copies of $R_{R}$ is a C3 module. (4) Every projective right $R$-module is a C2 (or C3) module. (5) $R$ is a right perfect ring and every finite direct sum copies of $R_{R}$ is a C2 (or C3) module. This shows that right countably $Σ$-C2 rings are just the rings whose right finitistic projective dimension r$FPD(R)$=sup\{$Pd_{R}(M)|$ $M$ is a right $R$-module with $Pd_{R}(M)<\infty$\}=0, which were introduced by Hyman Bass in 1960.

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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