On the decomposition and direct sums of modules (Q1804707)

A module \(M\) is called a CS-module (\(n\)-CS-module) if every closed submodule \(A\) of \(M\) (\(A\) of \(M\) with \(\text{U-}\dim(A)\leq n\)) is a direct summand. \(M\) is quasi-continuous (\(n\)-quasi-continuous) if it is a CS (\(n\)-CS) module and also satisfies the condition: for all \(X, Y \subset^ \oplus M\) (\(X, Y \subset^ \oplus M\) with \(\text{U-}\dim(X)\), \(\text{U-}\dim(Y)\leq n\)), where \(X\cap Y = 0\), then \(X \oplus Y \subset^ \oplus M\). The first part of the paper is devoted to a generalization of Matlis- Papp's theorem: \(R\) is right noetherian if and only if every injective module is a direct sum of indecomposable submodules. The author proves that over a right noetherian ring, every closed submodule of a 1-quasi- continuous module is a direct sum of uniform submodules. As a consequence, \(R\) is right noetherian if and only if every 1-quasi- continuous \(R\)-module is a direct sum of uniform submodules. In the second part, the author studies direct sums of CS (and 1-CS) modules and proves several interesting results, mainly concluding that a direct sum \(M = \oplus_{i \in I} M_ i\) of CS (1-CS) modules \(M_ i\) becomes a CS (\(n\)-CS) module under additional conditions imposed on the summands \(M_ i\). Some other results are provided for such a sum to be quasi- continuous under stronger conditions on the \(M_ i\).