Finitely-hollow lifting and hollow-weak lifting modules are two generalizations of hollow lifting module which is a weaker version of lifting module. In this paper, we have defined another generalization of hollow-lifting module namely finitelyhollow-weak lifting (f-hollow-weak lifting) module. Various properties of this module have been seen and found that, it is not closed under quotients and summands in general, so sufficient conditions are provided under which f-hollow-weak lifting property of modules is inherited by its quotients and summands. Finally, we have given some characterizations of f-hollow-weak lifting module.
In this work, [Formula: see text]-lifting modules, a special extension of [Formula: see text]-lifting which is a further generalization of lifting modules, are studied. Here, we observed that finite direct sum of [Formula: see text]-lifting modules may not be [Formula: see text]-lifting, so we provided various sufficient conditions for which [Formula: see text]-lifting modules are closed under direct sum. Moreover, we have introduced and studied the properties of a new version of [Formula: see text]-lifting module namely, [Formula: see text]-lifting, [Formula: see text]-lifting and completely [Formula: see text]-lifting modules and develop some more properties of the [Formula: see text]-lifting modules in terms of these modules. Further, we proved that if the direct sum of arbitrary family of hollow modules is [Formula: see text]-lifting, then the arbitrary direct sum of [Formula: see text]-lifting [Formula: see text]-module is again [Formula: see text]-lifting.
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