摘要
Let R be an associative ring with identity. An R-module M is called an NCS module if l（M）∩y（M） = {0}, where l（M） and y（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 ∑-CS if and only if R is right perfect and right countably ∑-NCS. Recall that a ring R is called right ∑-CS if every direct sum of copies of RR is a CS module. And a ring R is called right countably ∑-NCS if every direct sum of countable copies of RR is an NCS module.
Let R be an associative ring with identity. An R-module M is called an NCS module if l（M）∩y（M） = {0}, where l（M） and y（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 ∑-CS if and only if R is right perfect and right countably ∑-NCS. Recall that a ring R is called right ∑-CS if every direct sum of copies of RR is a CS module. And a ring R is called right countably ∑-NCS if every direct sum of countable copies of RR is an NCS module.
基金
Acknowledgements The authors would like to thank Professor Dinh Van Huynh and Professor Sergio Lopez-Permouth for their nice suggestions. This work was supported by the Natural Science Foundation of Jiangsu Province （Nos. BK20130599, 20141327）, the National Natural Science Foundation of China （Grant No. 11371089）, and the Project-sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.
