## On the number and ideal theory of one-dimensional noetherian integral domains. (Zur Zahlen- und Idealtheorie eindimensionaler noetherscher Integritätsbereiche.)(German)Zbl 0743.13001

Let $$R$$ be a noetherian domain, $$\alpha\in R$$, $$\alpha\neq 0$$, $$\alpha$$ not a unit, so $$\alpha$$ is a product of indecomposable elements. For every $$m\geq 2$$, let $$\mu_ m(R)$$ be the supremum of all $$k$$ such that there exist indecomposable elements $$u_ i$$, $$v_ j\in R$$, $$1\leq i\leq m$$, $$1\leq j\leq k$$, with $$u_ 1u_ 2\ldots u_ m=v_ 1v_ 2\ldots v_ k$$. Thus, $$m\leq \mu_ m(R)\leq\mu_{m+1}(R)\leq \infty$$. If $$\mu_ m(R)<\infty$$ for every $$m\geq 2$$, $$R$$ is said to be tame. — A. Geroldinger and G. Lettl have shown in 1972 the following results.
Theorem A. Every Krull domain with finite class number is tame.
Theorem B. Let $$R$$ be an order in a number field $$K$$ of degree $$d$$, having conductor $$F$$.
(i) Assume that for every prime ideal $$P$$ of $$R$$, containing $$F$$, the set $$\{Q\mid$$ invertible irreducible primary ideal, $$\sqrt Q=P\}$$ is finite. Then $$R$$ is tame.
(ii) If $$d=2$$, the hypothesis in (i) above is equivalent to: $$F\subseteq PR$$ for every rational prime $$P$$ which is decomposable in $$K$$.
In the present interesting paper, the author extends the above results to a wider framework: Let $$R$$ be a $$Z$$-ring, that is, a noetherian domain in which every invertible ideal is a product of primary ideals. For example, every one-dimensional noetherian domain (in particular, every order of a global field) is a $$Z$$-ring; also every factorial and even weakly factorial domain, is a $$Z$$-ring. Denote by $${\mathcal P}(R)$$ the set of all prime ideals of height 1 of $$R$$. Let $${\mathcal T}(R)$$ be the multiplicative monoid of invertible ideals of $$R$$. Let $${\mathcal T}_ 0(R)$$ be the set of all indecomposable elements of $${\mathcal T}(R)$$; these are the primary ideals of $$R$$ which are invertible and irreducible. — For each $$P\in{\mathcal P}(R)$$, let $$\Omega(P)$$ be the set of all $$Q\in{\mathcal T}_ 0(R)$$ such that $$\sqrt Q=P$$.
Theorem A is extended as follows: Theorem 1. Assume that $$R$$ is a $$Z$$- ring; $$\#\Omega(P)<\infty$$ for every $$P+{\mathcal P}(R)$$; $$\Omega(P)=\{P\}$$ for almost all $$P\in {\mathcal P}(R)$$; the Picard group of $$R$$ is finite. Then the ring $$R$$ is tame.
If $$R$$ is a domain, let $$\overline R$$ denote its integral closure (in the field of fractions), let $$R^*$$ denote the multiplicative group of units. Theorem $$B$$ is extended as follows: Theorem 2. Let $$R$$ be a noetherian domain of height 1. Assume that for every prime ideal $$P$$ of $$R$$, the ring $$\overline R_ P$$ is finitely generated $$R_ P$$-module. Then the following conditions are equivalent:
(a) $$\#\Omega(P)<\infty$$;
(b) There exists only one prime ideal $$\overline P$$ in $$\overline R$$, such that $$\overline P\cap R=P$$ and for $$\overline P$$, $$(\overline R_{\overline P}^* :R_ P)<\infty$$. The paper contains also various corollaries.

### MSC:

 13A05 Divisibility and factorizations in commutative rings 13E05 Commutative Noetherian rings and modules 13A10 Radical theory on commutative rings (MSC2000) 11A05 Multiplicative structure; Euclidean algorithm; greatest common divisors 13G05 Integral domains

### Keywords:

radicals; integral closure; group of units; noetherian domain
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### References:

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