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**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.

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.

Reviewer: P.Ribenboim (Kingston / Ontario)

### 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 |

Full Text:
DOI

### References:

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