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.

Key Words:

• Star-operation
• (Quasi-)*-Cancellation ideal
• *-Locally principal

2000 Mathematics Subject Classification:

• 13A15
• 13G05

ACKNOWLEDGMENTS

The author would like to thank the referee who suggested him the contents of Theorem 2.9 and Corollary 2.10.

Notes

Communicated by R. Wiegand.