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-AS, 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).