Co-coatomic lifting modules

The objective of this paper is to examine the lifting modules with the assumption that a submodule can be considered as a co-coatomic. A module [Formula: see text] is called a co-coatomic lifting module if, for every proper co-coatomic submodule [Formula: see text] of [Formula: see text], there exists a decomposition [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text]. Co-coatomic lifting modules are situated between lifting modules and cofinitely lifting modules. We demonstrate that if [Formula: see text] is a distributive co-coatomic lifting module, then [Formula: see text] is also co-coatomic lifting for any submodule [Formula: see text] of [Formula: see text]. Moreover, the co-coatomic direct summand of a co-coatomic lifting module is co-coatomic lifting. If every [Formula: see text]-module [Formula: see text] is co-coatomic lifting, then [Formula: see text] is semiperfect. Furthermore, we derive that a finitely generated co-coatomic lifting module is discrete under certain condition.