Let M be an R-module. If every essential submodule of M has a g-supplement in M, then M is called an essential g-supplemented (or briefly eg-supplemented) module. If every essential submodule has ample g-supplements in M, then M is called an amply essential g-supplemented (or briefly amply eg-supplemented) module. In this work, some properties of these modules are investigated. It is proved that every factor module and every homomorphic image of an amply eg-supplemented module are amply eg-supplemented. Let M be a projective and eg-supplemented module. Then every finitely M-generated R-module is amply eg-supplemented.

In this work oplus-e-supplemented modules are defined and some properties of these modules are investigated. It is proved that the finite direct sum of oplus-e-supplemented modules is also oplus-e-supplemented. Let M be a distributive and oplus-e-supplemented R-module. Then every factor module and homomorphic image of M are oplus-e-supplemented. Let M be a oplus-e-supplemented R-module with SSP property. Then every direct summand of M is oplus-e-supplemented.

In this work, strongly gradical supplemented modules are de…ned and some properties of these modules are investigated. Every ring has unity and every module is unital left module in this work. It is proved that every direct summand of a strongly gradical supplemented module is strongly gradical supplemented. Let f : M ! N be an Rmodule epimorphism and Ker (f) be a direct summand of M. If M is strongly gradical supplemented, then N is also strongly gradical supplemented. Let M be a strongly gradical supplemented Rmodule and K M. If V is a g-radical supplement submodule in M for every g-radical supplement submodule V =K in M=K, then M=K is strongly gradical supplemented.

In this work oplus-g-radical supplemented modules are defined and some properties of these modules are investigated. It is proved that the finite direct sum of oplus-g-radical supplemented modules is also oplus-g-radical supplemented.

In this work, we define Rad-oplus-supplemented and strongly Rad-oplus-supplemented lattices and give some properties of these lattices. We generalize some properties of Rad-oplus-supplemented modules to lattices.