ABSTRACT
This chapter introduces extending modules and provides elementary properties. It investigates injectivity properties and the relationship with modules of finite uniform dimension. Every semisimple module is extending, because every submodule is a direct summand. In addition, every uniform module is extending because every non-zero submodule is essential. The chapter discusses self-hereditary extending modules, and as the main result, it proves that any self-hereditary extending module over an arbitrary ring is a direct sum of noetherian uniform modules. Continuous and quasi-continuous modules were studied by various authors and a rich theory was developed. For a good account of this see the monograph by Mohamed and Muller which covers not only continuous and quasi-continuous modules but their duals discrete and quasi-discrete modules.
