On Krull domains

Abstract

One aim of this article is to provide for Krull domains a star-operation analogue of the following result: An integral domain D is a Dedekind domain if and only if each nonzero ideal A of D is strongly two generated. A nonzero ideal A of an integral domain D is called strongly two generated if for each x e A\{0} there is y e A such that A = xD + yD. Lantz and Martin show in [17] that a strongly two generated ideal is invertible. Following this lead we define a strongly *-type 2 ideal, for a star-operation *, as a nonzero ideal A such that for each x e A\{0}, there is y ~ A* such that (x, y)* = A*. Then in Section I we characterize Krult domains in terms of strongly *-type 2 ideals. Recently there has been considerable activity [1, 7, 12, 26] (some of it inspired by an earlier preprint version of the present paper) in characterizing a Krult domain in terms of the ,-invertibility of some or all fractional ideals of D. These results are interesting in that they indicate that most of the characterizations of Dedekind domains have *-oper- ation analogues for Krull domains. In Section 2 we continue this line of investigation by coordinating some of the recent results with some new characterizations of Krull domains in terms of *-invertibility.