Chain conditions on a generalization of supplemented modules

In this paper we show that the module M is Artinian if and only if M is a δ-GAS-module and satisfies DCC on generalized δ-supplemented submodules and on δ-small submodules and then we conclude that if M is a finitely generated δ-GAS-module, then M is Artinian if and only if M satisfies DCC on δ-small submodules. For a module M with ACC on δ-small submodules we show that if M is δ-GAS, then M is amply δ-supplemented and M satisfies δ − P^∗ if and only if M is δ-lifting. Finally we conclude that if M is a δ-GS (δ-GAS) module and δ(M) is Noetherian, then M is δ-supplemented (amply δ-supplemented)