中文摘要：

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 R _R is a CS module. And a ring R is called right countably Σ-NCS if every direct sum of countable copies of R _R is an NCS module.

英文摘要：

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.