
Generalization of CS condition
Liang SHEN, Wenxi LI
Front. Math. China ›› 2017, Vol. 12 ›› Issue (1) : 199-208.
Generalization of CS condition
Let R be an associative ring with identity. An R-module M is called an NCS module if , where and 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.
NCS modules / NCS rings / CS rings / Σ-CS rings / countably Σ-NCS
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