Relations Between SIP-Lifting and Coatomic Modules

This study investigates the relationship between SIP-lifting modules and coatomic modules in module theory. The SIP-lifting property extends the classical summand intersection property by allowing intersections of direct summands to contain a direct summand modulo a small submodule, thus preserving essential decomposition behavior. Coatomic modules are defined by the condition that every submodule is contained in a maximal submodule. We analyze how coatomicity interacts with SIP-lifting and identify conditions under which coatomic modules exhibit SIP-lifting behavior. Examples over matrix rings and semisimple Artinian rings are presented, together with new characterization and connection results.