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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

Citations:

Zbl 0655.13015
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References:

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