LetDbe an integral domain with quotient fieldK, let (F(D) (f(D)) be the set of nonzero (finitely generated) fractional ideals ofD, and let ★ be a star-operation onF(D).ForA ∈ F(D) andthere existsJ∈f(D)such thatJ★=D, andxJ⊆A}.ThenA★w= {x ∈K| existsJ ∈ f(D) such thatJ★=D, and xJ⊆A}. Thenand ★ware star-operations onF(D)that satisfy. Moreover,is the greatest (finite character) star-operation Δ ≤ ★ with (A∩B)Δ=AΔ∩BΔ.We also show that ★w-Max(D)= ★s-Max(D) andA★w=∩{AP|P∈★s-Max(D)}.LetL★w(D) = {A|Ais an integral ★w-ideal}∪{0}. ThenL★w(D) forms anr-lattice. IfDsatisfiesACCon integral ★w-ideals,L*w(D) is a Noether lattice and hence primary decomposition, the Krull intersection theorem, and the principal ideal theorem hold for *w-ideals ofD. For the case of ★=υ,★wis thew-operation introduced by Wang Fanggui and R.L. McCasland.