Generalization of CS condition

Liang SHEN, Wenxi LI

Front. Math. China ›› 2017, Vol. 12 ›› Issue (1) : 199-208.

PDF(147 KB)
PDF(147 KB)
Front. Math. China ›› 2017, Vol. 12 ›› Issue (1) : 199-208. DOI: 10.1007/s11464-016-0596-x
RESEARCH ARTICLE
RESEARCH ARTICLE

Generalization of CS condition

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Abstract

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.

Keywords

NCS modules / NCS rings / CS rings / Σ-CS rings / countably Σ-NCS

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Liang SHEN, Wenxi LI. Generalization of CS condition. Front. Math. China, 2017, 12(1): 199‒208 https://doi.org/10.1007/s11464-016-0596-x

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