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 $\mathcal R(H)=\mathsf {Ca}(H)=[2, \mathsf c(H)].$

MSC:

 13A05 Divisibility and factorizations in commutative rings 13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations 20M13 Arithmetic theory of semigroups
Full Text:

References:

 [1] N.R. Baeth and D. Smertnig, Factorization theory: From commutative to noncommutative settings, J. Algebra 441 (2015), 475–551. · Zbl 1331.20074 [2] P. Baginski, S.T. Chapman, C. Crutchfield, K.G. Kennedy and M. Wright, Elastic properties and prime elements, Result. Math. 49 (2006), 187–200. · Zbl 1115.20051 [3] P. Baginski, S.T. Chapman, M. Holden and T. Moore, Asymptotic elasticity in atomic monoids, Semigroup Forum 72 (2006), 134–142. · Zbl 1099.20033 [4] T. Barron, C. O’Neill and R. Pelayo, On the set of elasticities in numerical monoids, Semigroup Forum 94 (2017), 37–50. · Zbl 1380.20056 [5] G. Bhowmik and J.-C. Schlage-Puchta, Davenport’s constant for groups of the form $$\mathbb{Z}_3 ⊕ \mathbb{Z}_3 ⊕ \mathbb{Z}_{3d}$$, in Additive combinatorics, A. Granville, M.B. Nathanson and J. Solymosi, eds., American Mathematical Society, Providence, 2007. [6] C. Bowles, S.T. Chapman, N. Kaplan and D.Reiser, On delta sets of numerical monoids, J. Alg. Appl. 5 (2006), 695–718. · Zbl 1115.20052 [7] G.W. Chang, Every divisor class of Krull monoid domains contains a prime ideal, J. Algebra 336 (2011), 370–377. · Zbl 1244.13003 [8] S.T. Chapman, M. Holden and T. Moore, Full elasticity in atomic monoids and integral domains, Rocky Mountain J. Math. 36 (2006), 1437–1455. · Zbl 1152.20048 [9] S.T. Chapman, N. Kaplan, T. Lemburg, A. Niles and C. Zlogar, Shifts of generators and delta sets of numerical monoids, Inter. J. Alg. Comp. 24 (2014), 655–669. · Zbl 1315.20054 [10] F. Chen and S. Savchev, Long minimal zero-sum sequences in the groups $${C}_2^{r-1} ⊕ {C}_{2k}$$, Integers 14 (2014), paper A23. · Zbl 1308.11028 [11] S. Colton and N. Kaplan, The realization problem for delta sets of numerical monoids, J. Commutative Algebra 9 (2017), 313–339. · Zbl 1431.20039 [12] A. Geroldinger, Sets of lengths, Amer. Math. Month. 123 (2016), 960–988. · Zbl 1391.13004 [13] ——–, Additive group theory and non-unique factorizations, in Combinatorial number theory and additive group theory, A. Geroldinger and I. Ruzsa, eds., Advanced Courses in Mathematics, Birkhäuser, Berlin, 2009. · Zbl 1221.20045 [14] A. Geroldinger, D.J. Grynkiewicz, and W.A. Schmid, The catenary degree of Krull monoids I, J. Th. Nombres Bordeaux 23 (2011), 137–169. · Zbl 1253.11101 [15] A. Geroldinger and F. Halter-Koch, Non-unique factorizations. algebraic, combinatorial and analytic theory, Pure Appl. Math. 278 (2006). · Zbl 1113.11002 [16] A. Geroldinger and P. Yuan, The set of distances in Krull monoids, Bull. London Math. Soc. 44 (2012), 1203–1208. · Zbl 1255.13002 [17] A. Geroldinger and Q. Zhong, The catenary degree of Krull monoids II, J. Australian Math. Soc. 98 (2015), 324–354. · Zbl 1373.20074 [18] D.J. Grynkiewicz, Structural additive theory, Springer, New York, 2013. · Zbl 1368.11109 [19] F. Halter-Koch, Ideal systems, An introduction to multiplicative ideal theory, Marcel Dekker, New York, 1998. · Zbl 0953.13001 [20] C. O’Neill, V. Ponomarenko, R. Tate and G. Webb, On the set of catenary degrees of finitely generated cancellative commutative monoids, Int. J. Alg. Comp. 26 (2016), 565–576. · Zbl 1357.20027 [21] W.A. Schmid, The inverse problem associated to the Davenport constant for $${C}_2 ⊕ {C}_2 ⊕ {C}_{2n}$$, and applications to the arithmetical characterization of class groups, Electr. J. Comb. 18 (2011). · Zbl 1244.11088 [22] D. Smertnig, Sets of lengths in maximal orders in central simple algebras, J. Algebra 390 (2013), 1–43. · Zbl 1295.16023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.