Generalizations of basic and large submodules of QT AG-modules

A QT AG-module M over an associative ring R with unity is k-projective if H-k (M) = 0 and for a limit ordinal sigma, it is sigma-projective if there exists a submodule N bounded by sigma such that M/N is a direct sum of uniserial modules. M is totally projective if it is sigma-projective for all limit ordinals sigma. If alpha denotes the class of all QT AG-modules M such that M/H-beta (M) is totally projective for every ordinal beta < alpha, then these modules are called alpha-modules. Here we study these alpha-modules and generalize the concept of basic submodules as alpha-basic submodules. It is found that every alpha-module M contains an alpha-basic submodule and any two alpha-basic submodules of M are isomorphic. A submodule L of an alpha-module is alpha-large if M = L + B, for any alpha-basic submodule B of M. Many other interesting properties of alpha-basic, alpha-large and alpha-modules are studied.