##
**Minimal relations and catenary degrees in Krull monoids.**
*(English)*
Zbl 1409.13004

Let \(H\) be a Krull monoid with class group \(G\), let \(H_{\text{red}}\) be the reduced monoid of \(H\), and let \(\mathcal A(H_{\text{red}})\) be the set of atoms of \(H_{\text{red}}\). The free abelian monoid \(\mathsf Z (H) = \mathcal F (\mathcal A (H_{\text{red}}))\) is the factorization monoid of \(H\), and the unique homomorphism \(\pi : \mathsf Z (H) \to H_{\text{red}}\) satisfying \(\pi (u) = u\) for all \(u \in \mathcal A (H_{\text{red}})\) is the factorization homomorphism of \(H\) (so \(\pi\) maps a formal product of atoms onto its product in \(H_{\text{red}}\)). For \(a \in H\), \(\mathsf Z_H (a) = \mathsf Z (a) = \pi^{-1} (a) \subset \mathsf Z (H)\) is the set of factorizations of \(a\). Then \(H\) is atomic (i.e., each non-unit can be written as a finite product of atoms) if and only if \(\mathsf Z (a) \ne \emptyset\) for each \(a \in H\), and \(H\) is factorial if and only if \(|\mathsf Z (a)|=1\) for each \(a \in H\).

Two factorizations \(z,\, z' \in \mathsf Z (H)\) can be written in the form \[ z = u_1 \cdot \ldots \cdot u_lv_1 \cdot \ldots \cdot v_m \quad \text{and} \quad z' = u_1 \cdot \ldots \cdot u_lw_1 \cdot \ldots \cdot w_n \] with \[ \{v_1 ,\ldots, v_m \} \cap \{w_1, \ldots, w_n \} = \emptyset, \] where \(l,\,m,\, n\in \mathbb N_0\) and \(u_1, \ldots, u_l,\,v_1, \ldots,v_m,\, w_1, \ldots, w_n \in \mathcal A(H_{\text{red}})\). Then \(\gcd (z, z') = u_1 \cdot \ldots \cdot u_l\), and we call \(\mathsf d (z, z') = \max \{m,\, n\} = \max \{ |z \gcd (z, z')^{-1}|, |z'\gcd (z, z')^{-1}| \}\in \mathbb N_0\) the distance between \(z\) and \(z'\). It is easy to verify that \(\mathsf d : \mathsf Z (H) \times \mathsf Z (H) \to \mathbb N_0\) has all the usual properties of a metric.

Let \(a \in H\) and \(N \in \mathbb N_0 \cup \{\infty\}\). A finite sequence \(z_0, \ldots, z_k \in \mathsf Z (a)\) is called an \(N\)-chain of factorizations if \(\mathsf d (z_{i-1}, z_i) \le N\) for all \(i \in [1, k]\). We denote by \(\mathsf c (a)\) the smallest \(N \in \mathbb N _0 \cup \{\infty\}\) such that any two factorizations \(z,\, z' \in \mathsf Z (a)\) can be concatenated by an \(N\)-chain and call \[ \mathsf c(H) = \sup \{ \mathsf c(b) \, \mid b \in H\} \in \mathbb N_0 \cup \{\infty\} \] the catenary of \(H\).

The set of catenary degrees is defined as \[ \mathsf {Ca}(H)=\{\mathsf c(a)\mid a\in H, \mathsf c(a)>0\} \] and the set of distances in minimal relations \(\mathcal {R}(H)\) is defined to be all \(d\in \mathbb N\) having the following property: There is an element \(a\in H\) having two distinct factorizations \(z\) and \(z'\) with distance \(\mathsf d(z,z') = d\), but there is no \(d-1\)-chain of factorizations concatenating \(z\) and \(z'\).

The authors prove that every finite nonempty subset of \(\mathbb N_{\ge 2}\) can be realized as the set of catenary degrees of a Krull monoid with finite class group. Under the assumption that every class contains a prime divisor (for example, a ring of integers in an algebraic number field or a holomorphy ring in an algebraic function field) and a reasonable condition on the Davenport constant of \(G\), they also show that \[ \mathcal R(H)=\mathsf {Ca}(H)=[2, \mathsf c(H)]. \]

Two factorizations \(z,\, z' \in \mathsf Z (H)\) can be written in the form \[ z = u_1 \cdot \ldots \cdot u_lv_1 \cdot \ldots \cdot v_m \quad \text{and} \quad z' = u_1 \cdot \ldots \cdot u_lw_1 \cdot \ldots \cdot w_n \] with \[ \{v_1 ,\ldots, v_m \} \cap \{w_1, \ldots, w_n \} = \emptyset, \] where \(l,\,m,\, n\in \mathbb N_0\) and \(u_1, \ldots, u_l,\,v_1, \ldots,v_m,\, w_1, \ldots, w_n \in \mathcal A(H_{\text{red}})\). Then \(\gcd (z, z') = u_1 \cdot \ldots \cdot u_l\), and we call \(\mathsf d (z, z') = \max \{m,\, n\} = \max \{ |z \gcd (z, z')^{-1}|, |z'\gcd (z, z')^{-1}| \}\in \mathbb N_0\) the distance between \(z\) and \(z'\). It is easy to verify that \(\mathsf d : \mathsf Z (H) \times \mathsf Z (H) \to \mathbb N_0\) has all the usual properties of a metric.

Let \(a \in H\) and \(N \in \mathbb N_0 \cup \{\infty\}\). A finite sequence \(z_0, \ldots, z_k \in \mathsf Z (a)\) is called an \(N\)-chain of factorizations if \(\mathsf d (z_{i-1}, z_i) \le N\) for all \(i \in [1, k]\). We denote by \(\mathsf c (a)\) the smallest \(N \in \mathbb N _0 \cup \{\infty\}\) such that any two factorizations \(z,\, z' \in \mathsf Z (a)\) can be concatenated by an \(N\)-chain and call \[ \mathsf c(H) = \sup \{ \mathsf c(b) \, \mid b \in H\} \in \mathbb N_0 \cup \{\infty\} \] the catenary of \(H\).

The set of catenary degrees is defined as \[ \mathsf {Ca}(H)=\{\mathsf c(a)\mid a\in H, \mathsf c(a)>0\} \] and the set of distances in minimal relations \(\mathcal {R}(H)\) is defined to be all \(d\in \mathbb N\) having the following property: There is an element \(a\in H\) having two distinct factorizations \(z\) and \(z'\) with distance \(\mathsf d(z,z') = d\), but there is no \(d-1\)-chain of factorizations concatenating \(z\) and \(z'\).

The authors prove that every finite nonempty subset of \(\mathbb N_{\ge 2}\) can be realized as the set of catenary degrees of a Krull monoid with finite class group. Under the assumption that every class contains a prime divisor (for example, a ring of integers in an algebraic number field or a holomorphy ring in an algebraic function field) and a reasonable condition on the Davenport constant of \(G\), they also show that \[ \mathcal R(H)=\mathsf {Ca}(H)=[2, \mathsf c(H)]. \]

Reviewer: Qinghai Zhong (Graz)

### MSC:

13A05 | Divisibility and factorizations in commutative rings |

13F05 | Dedekind, Prüfer, Krull and Mori rings and their generalizations |

20M13 | Arithmetic theory of semigroups |

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\textit{Y. Fan} and \textit{A. Geroldinger}, J. Commut. Algebra 11, No. 1, 29--47 (2019; Zbl 1409.13004)

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