Isomorphic minimal direct summands of QTAG-modules

Generalizing the classical concept of an h-purifiable submodule, we introduce the notion of a quasi h-purifiable submodule and give a systematic study of their basic and specific properties. Some characterizations of these objects are obtained, and in particular, we prove that an h-reduced QTAG-module M is closed if and only if all quasi h-purifiable submodules of M are summandable submodules. We establish another characterization of closed modules; namely, in a closed QTAG-module M, all minimal direct summands containing a summandable submodule are isomorphic. In addition, we also examine these considerations to the class of quasi-complete QTAG-modules.