A generalization of singular modules in terms of purely extending property (Q6655197)

The purpose of this article is to study and generalise the following known classes of modules. A submodule \(N\) of a module \(M\) is \textit{closed} if it has no essential extension contained in \(M\); \(M\) is \textit{extending} if every closed submodule is a direct summand and \textit{purely extending} if every closed pure submodule is a direct summand. \(N\) is \textit{\(c\)-closed} if whenever \(N\leq B\leq M\) and \(B/N\) is singular, then \( N = B\). If each \(c\)-closed submodule of a module \(M\) is is a direct summand, then \(M\) is called \textit{\(CCLS\)}. A module \(M\) is \textit{purely \(CCLS\)} if each \(c\)-closed submodule is pure in \(M\).\N\NThe authors determine relations among \(c\)-closed, \(CCLS\) and purely extending modules and present several characterisations and properties of purely \(CCLS\) modules. For example, they present conditions for a direct sum of purely \(CCLS\) modules to be purely \(CCLS\).