On Direct Sums of Lifting Modules and Internal Exchange Property #

Abstract

ABSTRACT Harada (cf. 1983) introduced the lifting property for maximal submodules and the extending property for simple submodules, and Oshiro (cf. 1983b) definitely introduced lifting modules and extending modules. Since then, the method and results in their works urged not only the study on (quasi-)discrete and (quasi-)continuous modules but also the one on themselves. The reader is referred to Mohamed and Müller (1990) and Dung et al. (1994) for the research on lifting modules and extending modules. In this article, we study the following fundamental open problems: When is a direct sum ⨁IMi of lifting (extending) modules {Mi}I lifting (extending)? These problems are unsolved even in the case that the index set I is finite. Now, the purpose of this article is to study the problem concerned with finite direct sums of lifting modules.