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Alfred Geroldinger; Qinghai Zhong
A characterization of class groups via sets of lengths II
Journal de théorie des nombres de Bordeaux, 29 no. 2 (2017), p. 327-346, doi: 10.5802/jtnb.983
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Class. Math.: 11B30, 11R27, 13A05, 13F05, 20M13
Keywords: Krull monoids, maximal orders, seminormal orders, class groups, arithmetical characterizations, sets of lengths, zero-sum sequences, Davenport constant

Résumé - Abstract

Let $H$ be a Krull monoid with finite class group $G$ and suppose that every class contains a prime divisor. If an element $a \in H$ has a factorization $a=u_1 \cdot \ldots \cdot u_k$ into irreducible elements $u_1, \ldots , u_k \in H$, then $k$ is called the length of the factorization and the set $\mathsf {L} (a)$ of all possible factorization lengths is the set of lengths of $a$. It is classical that the system $\mathcal{L} (H) = \lbrace \mathsf {L} (a) \mid a \in H \rbrace$ of all sets of lengths depends only on the class group $G$, and a standing conjecture states that conversely the system $\mathcal{L} (H)$ is characteristic for the class group. We verify the conjecture if the class group is isomorphic to $C_n^r$ with $r,n \ge 2$ and $r \le \max \lbrace 2, (n+2)/6\rbrace$. Indeed, let $H^{\prime }$ be a further Krull monoid with class group $G^{\prime }$ such that every class contains a prime divisor and suppose that $\mathcal{L} (H)= \mathcal{L} (H^{\prime })$. We prove that, if one of the groups $G$ and $G^{\prime }$ is isomorphic to $C_n^r$ with $r,n$ as above, then $G$ and $G^{\prime }$ are isomorphic (apart from two well-known pairings).

Bibliography

[1] Nicholas R. Baeth & Alfred Geroldinger, Monoids of modules and arithmetic of direct-sum decompositions, Pac. J. Math. 271 (2014), p. 257-319 Article
[2] Nicholas R. Baeth & Daniel Smertnig, Factorization theory: From commutative to noncommutative settings, J. Algebra 441 (2015), p. 475-551 Article
[3] Paul Baginski, Alfred Geroldinger, David J. Grynkiewicz & Andreas Philipp, Products of two atoms in Krull monoids and arithmetical characterizations of class groups, Eur. J. Comb. 34 (2013), p. 1244-1268 Article
[4] Gyu Whan Chang, Every divisor class of Krull monoid domains contains a prime ideal, J. Algebra 336 (2011), p. 370-377 Article
[5] Scott T. Chapman, Wolfgang A. Schmid & William W. Smith, On minimal distances in Krull monoids with infinite class group, Bull. Lond. Math. Soc. 40 (2008), p. 613-618 Article
[6] Alberto Facchini, Krull monoids and their application in module theory, Algebras, Rings and their Representations, World Scientific, 2006, p. 53–71
[7] Alfred Geroldinger, David J. Grynkiewicz & Wolfgang A. Schmid, The catenary degree of Krull monoids I, J. Théor. Nombres Bordx. 23 (2011), p. 137-169 Article
[8] Alfred Geroldinger & Franz Halter-Koch, Non-Unique Factorizations. Algebraic, Combinatorial and Analytic Theory, Pure and Applied Mathematics 278, Chapman & Hall/CRC, 2006
[9] Alfred Geroldinger & Yahya Ould Hamidoune, Zero-sumfree sequences in cyclic groups and some arithmetical application, J. Théor. Nombres Bordx. 14 (2002), p. 221-239 Article
[10] Alfred Geroldinger, Florian Kainrath & Andreas Reinhart, Arithmetic of seminormal weakly Krull monoids and domains, J. Algebra 444 (2015), p. 201-245 Article
[11] Alfred Geroldinger & Imre Z. Ruzsa, Combinatorial Number Theory and Additive Group Theory, Advanced Courses in Mathematics - CRM Barcelona, Birkhäuser, 2009, With a foreword by Javier Cilleruelo, Marc Noy and Oriol Serra (Coordinators of the DocCourse)
[12] Alfred Geroldinger & Wolfgang A. Schmid, “A characterization of class groups via sets of lengths”, http://arxiv.org/abs/1503.04679, 2015
[13] Alfred Geroldinger & Wolfgang A. Schmid, The system of sets of lengths in Krull monoids under set addition, Rev. Mat. Iberoam. 32 (2016), p. 571-588 Article
[14] Alfred Geroldinger & Rudolf Schneider, On Davenport’s constant, J. Comb. Theory 61 (1992), p. 147-152 Article
[15] Alfred Geroldinger & Qinghai Zhong, The set of minimal distances in Krull monoids, Acta Arith. 173 (2016), p. 97-120  MR 3503061
[16] David J. Grynkiewicz, Structural Additive Theory, Developments in Mathematics 30, Springer, 2013
[17] Hwankoo Kim & Young Soo Park, Krull domains of generalized power series, J. Algebra 237 (2001), p. 292-301 Article
[18] Alain Plagne & Wolfgang A. Schmid, “On congruence half-factorial Krull monoids with cyclic class group”, submitted
[19] Alain Plagne & Wolfgang A. Schmid, On the maximal cardinality of half-factorial sets in cyclic groups, Math. Ann. 333 (2005), p. 759-785 Article
[20] Wolfgang A. Schmid, Differences in sets of lengths of Krull monoids with finite class group, J. Théor. Nombres Bordx. 17 (2005), p. 323-345 Article
[21] Wolfgang A. Schmid, Arithmetical characterization of class groups of the form $\mathbb{Z} /n \mathbb{Z} \oplus \mathbb{Z} /n \mathbb{Z}$ via the system of sets of lengths, Abh. Math. Semin. Univ. Hamb. 79 (2009), p. 25-35 Article
[22] Wolfgang A. Schmid, Characterization of class groups of Krull monoids via their systems of sets of lengths a status report, in Number theory and applications. Proceedings of the international conferences on number theory and cryptography, Allahabad, India, December 2006 and February 2007, Hindustan Book Agency, 2009, p. 189-212
[23] Wolfgang A. Schmid, The inverse problem associated to the Davenport constant for ${C}_2 \oplus {C}_2 \oplus {C}_{2n}$, and applications to the arithmetical characterization of class groups, Electron. J. Comb. 18 (2011)
[24] Daniel Smertnig, Sets of lengths in maximal orders in central simple algebras, J. Algebra 390 (2013), p. 1-43 Article