Amply essential g-supplemented modules

. 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.