Principally ss-Supplemented Modules (Baghdad Science Journal, 2024)
In this paper, we introduce and study the concepts of principally ss-supplemented and principally ss-lifting modules. These two concepts are natural generalizations of the concepts of ss-supplemented and ss-lifting modules. Several properties of these modules are proven. Here, principally ss-lifting modules are focused on. New characterizations of principally ss-supplemented modules are made using principally ss-lifting modules. Here, weakly principally ss-supplemented is defined. It is proved that   a module T is weakly principally ss-supplemented module if and only if it is principally ss-supplemented. One of the first results states that every strongly local module is principally ss-supplemented. It is shown that if T be a hollow module, then T is principally ss-supplemented if and only if it is strongly local. If Rad(T) small in T, then T is principally ss-supplemented if and only if T is principally supplemented and Rad(T)⊆Soc(T). Moreover, if T=T_1⨁T_2 with T_1 and T_2 principally ss-supplemented modules and T is a duo, then T is principally ss-supplemented. It is also shown that, if T is indecomposable, then T is principally ss-lifting if and only if T is a principally hollow module besides if T is a principally hollow module then T is principally ss-supplemented.  In this work, the following results are proved: if T be a module with the property (ss -PD_1), then every indecomposable cyclic submodule of T is either small in T or a summand of T. Also, if T is a module over a local ring R and T has the property (ss-PD_1), then every cyclic submodule of T is either small in T, or a summand of T.