Generalization of CS condition
详细信息 查看全文
- 作者：Liang Shen ; Wenxi Li
- 关键词：NCS modules ; NCS rings ; CS rings ; Σ ; CS rings ; countably Σ ; NCS
- 刊名：Frontiers of Mathematics in China
- 出版年：2017
- 出版时间：February 2017
- 年：2017
- 卷：12
- 期：1
- 页码：199-208
- 全文大小：153 KB
- 刊物类别：Mathematics and Statistics
- 刊物主题：Mathematics
Mathematics
Chinese Library of Science - 出版者：Higher Education Press, co-published with Springer-Verlag GmbH
- ISSN：1673-3576
- 卷排序：12
文摘
Let R be an associative ring with identity. An R-module M is called an NCS module if C (M)∩S(M) = {0}, where C (M) and S(M) denote the set of all closed submodules and the set of all small submodules of M, respectively. It is clear that the NCS condition is a generalization of the well-known CS condition. Properties of the NCS conditions of modules and rings are explored in this article. In the end, it is proved that a ring R is right Σ-CS if and only if R is right perfect and right countably Σ-NCS. Recall that a ring R is called right Σ-CS if every direct sum of copies of RR is a CS module. And a ring R is called right countably Σ-NCS if every direct sum of countable copies of RR is an NCS module.