Characterizations of *-Cancellation Ideals of an Integral Domain

Abstract

Let D be an integral domain and * a star-operation on D. For a nonzero ideal I of D, let I * f = ⋃{J* | (0) ≠ J ⊆ I is finitely generated} and I * w = ⋂ P∈* f -Max(D) ID P . A nonzero ideal I of D is called a *-cancellation ideal if (IA)* = (IB)* for nonzero ideals A and B of D implies A* =B*. Let X be an indeterminate over D and N * = {f ∈ D[X] | (c(f))* =D}. We show that I is a * w -cancellation ideal if and only if I is * f -locally principal, if and only if ID[X] N * is a cancellation ideal. As a corollary, we have that each nonzero ideal of D is a * w -cancellation ideal if and only if D P is a principal ideal domain for all P ∈ * f -Max(D), if and only if D[X] N * is an almost Dedekind domain. We also show that if I is a * w -cancellation ideal of D, then I * w = I * f = I t , and I is * w -invertible if and only if I * w = J v for a nonzero finitely generated ideal J of D.