Goldie-SS-supplemented modules

In this study, the concept of Goldie-ss-supplemented modules is given with Tβ_ss^* W, notation and for T and W submodules of module X, ((T+W))⁄T⊆ (〖(Soc〗_s (X)+T))⁄T and ((T+W))⁄W⊆((〖Soc〗_s (X)+W))⁄W is an equivalance relation determined with the help of the relation β_ss^*, satisfying the conditions. Using this relation, Goldie-ss-supplemented modules are defined and several properties are examined. The application of epimorphism γ:X⟶Z and relation β_ss^* has been examined in maximal submodules, which provides relation β_ss^* under certain conditions. In addition, the basic properties and generalizations of the concept of Goldie-ss-lifting modules are determined with the help of the relation β_ss^*. With the help of this relation, Goldie-ss-lifting modules were also defined and their important properties were determined.