In this paper, considering L being a completely distributive lattice, we first introduce the concept of L -fuzzy ideal degrees in an effect algebra E , in symbol Dei . Further, we characterize L -fuzzy ideal degrees by cut sets. Then it is shown that an L -fuzzy subset A in E is an L -fuzzy ideal if and only if Dei(A)=⊤, which can be seen as a generalization of fuzzy ideals. Later, we discuss the relations between L -fuzzy ideals and cut sets (Lβ -nested sets and Lα -nested sets). Finally, we obtain that the L -fuzzy ideal degree is an (L,L) -fuzzy convexity. The morphism between two effect algebras is an (L,L) -fuzzy convexity-preserving mapping.