Generalization of CS condition

Abstract

Let R be an associative ring with identity. An R-module M is called an NCS module if C(M)\cap S(M)=\{\mathrm{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.

Keywords

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

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