## On Krull domains.(English)Zbl 0777.13011

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 $$x\in A\backslash\{0\}$$ there is $$y\in A$$ such that $$A=xD+yD$$. D. C. Lantz and M. R. Martin showed in Commun. Algebra 16, No. 9, 1759-1777 (1988; Zbl 0655.13015) 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 $$x\in A\backslash\{0\}$$, there is $$y\in A^*$$ such that $$(x,y)^*=A^*$$. Then in section 1 we characterize Krull domains in terms of strongly $$*$$-type 2 ideals. Recently there has been considerable activity in characterizing a Krull 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 $$*$$-operation 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.

### MSC:

 13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations 13A15 Ideals and multiplicative ideal theory in commutative rings 13E15 Commutative rings and modules of finite generation or presentation; number of generators

### Keywords:

Krull domains; star-operation; strongly two generated ideal

Zbl 0655.13015
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### References:

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