Direct sum of various generalizations of h-lifting modules

Abstract

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.