Two star-operations and their induced lattices

Abstract

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 =∩{A_P | 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.