St-closed Submodules and Semi-extending Modules

The main purpose of this work is to determine and study two new concepts (up to our knowledge) which are St-closed submodules and semi-extending modules. St-closed submodules is contained properly in the class of closed submodules where a submodule N of M is called St-closed in M if N has no proper semi-essential extension in M i.e if there exists a submodule K of M such that N is a semi-essential submodule of K then N = K. We investigate the main properties of this type of submodules and discuss some results that are useful in our work. The class of semi-extending modules is a generalization to the notion of extending modules where an R-module M is called semi-extending if every submodule of M is a semi-essential in a direct summand of M. Various properties of semi-extending modules are obtained and we study the relationships between this class of modules and other related concepts.