Mutually ss-supplemented modules

In this study, the notion of mutually ss-supplemented modules is defined by using the notion of ss-supplemented modules and it is obtained several properties of these modules. The concepts of amply mutual ss-supplement and amply mutual ss-supplemented modules are brought to the literature. Equivalent conditions of the mutual ss-supplemented submodule concept are explained. It is shown that for an S-module C with Rad(C) ⊆ Des(C) each submodule mutual supplement, is mutually ss-supplemented. It is mentioned that each semisimple module is mutually ss-supplemented using semisimple modules. The conditions under which a projective module can be mutually ss-supplemented are investigated. It is shown that a module C, each submodule a fully invariant submodule, is mutually ss-supplemented. Based on the concept of a refinable module, it has been proven that if a module C is an ss-supplemented module, C has direct summand such that they are mutually ss-supplemented. It is shown that the direct summand of a distributive mutually ss-supplemented module are also a mutually ss-supplemented module.