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On principally lifting modules. (English) Zbl 1128.16002

Discrete (quasi-discrete) modules form an important class in module theory. The class of lifting modules is obtained by considering only one of the defining conditions of quasi-discrete modules, namely the condition (D\(_1\)). Here the authors focus on and study principally lifting modules, or modules with the condition (PD\(_1\)). These modules are generalizations of lifting modules. The authors also study direct sums of P-hollow (semi-hollow) modules. They introduce the definition of relative P-projectivity, which is essential to examine direct sums of hollow, and of P-hollow (semi-hollow), modules for being principally lifting. Quasi-discrete modules are always direct sums of hollow submodules, the authors show that finite dimensional modules with the condition (PD\(_1\)) are direct sums of P-hollow (semi-hollow) submodules. In addition, the authors also obtain some properties for modules with (PD\(_1\)), which are in analogy with the known properties for lifting modules.

MSC:

16D40 Free, projective, and flat modules and ideals in associative algebras
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
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