On Supplementation and Generalized Projective Modules

Discrete (quasi) modules form an important class in module theory, they are studied extensively by many authors. The decomposition theorem for quasidiscrete modules plays an important rule in the better understanding of such modules. In fact, every quasidiscrete module is a direct sum of hollow submodules. Here we introduce some new concepts (weak quasidiscrete, and $S_{1}$- and $S_{2}$-supplemented modules) which generalize the concept of quasidiscrete modules. We show that some of the properties of quasidiscrete modules still hold in the class of weak quasidiscrete modules. We also obtain some properties for weak quasidiscrete modules, which are similar to the properties known for quasidiscrete modules. We introduce the concept of generalized relative projectivity (relative $S$-projective modules), and use it to characterize direct sums of hollow modules. In fact, relative $S$-projectivity is an essential condition for direct sums of hollow modules to be weak quasidiscrete modules.