Let D be an integral domain with quotient field K, let (F(D) (f(D)) be the set of nonzero (finitely generated) fractional ideals of D, and let ★ be a star-operation on F(D).For A ∈ F(D) and there exists J∈f(D) such that J★=D, and xJ ⊆ A}.Then A ★w = {x ∈ K | exists J ∈ f(D) such that J ★ = D, and xJ ⊆ A}. Then and ★w are star-operations on F(D) that satisfy . Moreover, is the greatest (finite character) star-operation Δ ≤ ★ with (A ∩B)Δ=A Δ∩ B Δ.We also show that ★ w -Max(D)= ★ s -Max(D) and A ★w =∩{AP | P ∈★ s -Max(D)}.Let L ★w (D) = {A | A is an integral ★ w -ideal}∪{0}. Then L ★w (D) forms an r-lattice. If D satisfies ACC on 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 of D. For the case of ★=υ,★ w is the w-operation introduced by Wang Fanggui and R.L. McCasland.
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