A generalization of singular modules in terms of purely extending property

In this article, we present and examine a type of modules that represent appropriate generalizations of each of the singular modules, purely extending and CCLS modules. A module M is called purely CCLS module if every c-closed submodule of M is pure in M. We locate purely CCLS module with generalizations of extending modules. We present some characterizations of purely CCLS and we show that the direct summand of purely CCLS modules is again purely CCLS. Furthermore, we go over when a direct sum of purely CCLS is likewise purely CCLS. Additionally, we provide adequate circumstances under which the direct sum of purely CCLS is purely CCLS.