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Sets of minimal distances and characterizations of class groups of Krull monoids. (English) Zbl 1426.11005

Summary: Let \(H\) be a Krull monoid with finite class group \(G\) such that every class contains a prime divisor. Then every non-unit \(a \in H\) can be written as a finite product of atoms, say \(a=u_1 \cdot \ldots \cdot u_k\). The set \(\mathsf L (a)\) of all possible factorization lengths \(k\) is called the set of lengths of \(a\). There is a constant \(M \in \mathbb N\) such that all sets of lengths are almost arithmetical multiprogressions with bound \(M\) and with difference \(d \in \Delta^\ast (H)\), where \(\Delta ^\ast (H)\) denotes the set of minimal distances of \(H\). We study the structure of \(\Delta ^\ast (H)\) and establish a characterization when \(\Delta ^\ast(H)\) is an interval. The system \(\mathcal L (H) = \{\mathsf L (a) \mid a \in H \}\) 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 confirm this conjecture (among others) if the class group is isomorphic to \(C_n^r\) with \(r,n \in \mathbb N\) and \(\Delta ^\ast(H)\) is not an interval.

MSC:

11B30 Arithmetic combinatorics; higher degree uniformity
11R27 Units and factorization
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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[1] Baeth, N.R., Geroldinger, A.: Monoids of modules and arithmetic of direct-sum decompositions. Pac. J. Math. 271, 257-319 (2014) · Zbl 1347.16006 · doi:10.2140/pjm.2014.271.257
[2] Chang, G.W.: Every divisor class of Krull monoid domains contains a prime ideal. J. Algebr. 336, 370-377 (2011) · Zbl 1244.13003 · doi:10.1016/j.jalgebra.2011.03.015
[3] Chapman, S.T., Fontana, M., Geroldinger, A., Olberding, B. (eds.): Multiplicative Ideal Theory and Factorization Theory. Proceedings in Mathematics and Statistics, vol. 170. Springer, Berlin (2016) · Zbl 1346.13002
[4] Chapman, S.T., Schmid, W.A., Smith, W.W.: On minimal distances in Krull monoids with infinite class group. Bull. Lond. Math. Soc. 40, 613-618 (2008) · Zbl 1198.20049 · doi:10.1112/blms/bdn040
[5] Facchini, A.; Facchini, A. (ed.); Fuller, K. (ed.); Ringel, CM (ed.); Santa-Clara, C. (ed.), Krull monoids and their application in module theory, 53-71 (2006), Hackensack · Zbl 1113.20049 · doi:10.1142/9789812774552_0006
[6] Geroldinger, A.: Sets of lengths. arXiv:1509.07462 · Zbl 0882.20039
[7] Geroldinger, A., Grynkiewicz, D.J., Schmid, W.A.: The catenary degree of Krull monoids I. J. Théor. Nombres Bordx. 23, 137-169 (2011) · Zbl 1253.11101 · doi:10.5802/jtnb.754
[8] Geroldinger, A., Halter-Koch, F.: Non-unique factorizations. Algebraic, combinatorial and analytic theory. In: Pure and Applied Mathematics, vol. 278. Chapman & Hall/CRC, Boca Raton (2006) · Zbl 1113.11002
[9] Geroldinger, A., Hamidoune, Y.O.: Zero-sumfree sequences in cyclic groups and some arithmetical application. J. Théor. Nombres Bordx. 14, 221-239 (2002) · Zbl 1018.11011 · doi:10.5802/jtnb.355
[10] Geroldinger, A., Ruzsa, I.: Combinatorial number theory and additive group theory. In: Advanced Courses in Mathematics—CRM Barcelona. Birkhäuser, Basel (2009) · Zbl 1177.11005
[11] Geroldinger, A., Schmid, W.A.: A characterization of class groups via sets of lengths. arXiv:1503.04679 · Zbl 1396.20069
[12] Geroldinger, A., Schmid, W.A.: The system of sets of lengths in Krull monoids under set addition. Rev. Mat. Iberoam. 32, 571-588 (2016) · Zbl 1417.11007 · doi:10.4171/RMI/895
[13] Geroldinger, A., Yuan, P.: The set of distances in Krull monoids. Bull. Lond. Math. Soc. 44, 1203-1208 (2012) · Zbl 1255.13002 · doi:10.1112/blms/bds046
[14] Geroldinger, A., Zhong, Q.: The catenary degree of Krull monoids II. J. Aust. Math. Soc. 98, 324-354 (2015) · Zbl 1373.20074 · doi:10.1017/S1446788714000585
[15] Geroldinger, A., Zhong, Q.: The set of minimal distances in Krull monoids. Acta Arith. 173, 97-120 (2016) · Zbl 1360.20053
[16] Geroldinger, A., Zhong, Q.: A characterization of class groups via sets of lengths II, J. Théor. Nombres Bordx. (to appear) · Zbl 1410.11121
[17] Grynkiewicz, D.J.: Structural Additive Theory. Developments in Mathematics. Springer, New York (2013) · Zbl 1368.11109 · doi:10.1007/978-3-319-00416-7
[18] Halter-Koch, F.: Ideal Systems. An Introduction to Multiplicative Ideal Theory. Marcel Dekker, New York (1998) · Zbl 0953.13001
[19] Kim, H., Park, Y.S.: Krull domains of generalized power series. J. Algebr. 237, 292-301 (2001) · Zbl 1039.13012 · doi:10.1006/jabr.2000.8581
[20] Plagne, A., Schmid, W.A.: On congruence half-factorial Krull monoids with cyclic class group (submitted) · Zbl 1456.20015
[21] Schmid, W.A.: Differences in sets of lengths of Krull monoids with finite class group. J. Théor. Nombres Bordx. 17, 323-345 (2005) · Zbl 1090.20034 · doi:10.5802/jtnb.493
[22] Schmid, W.A.: Arithmetical characterization of class groups of the form \[\mathbb{Z}/n \mathbb{Z} \oplus \mathbb{Z} /n \mathbb{Z}\] Z/nZ⊕Z/nZ via the system of sets of lengths. Abh. Math. Semin. Univ. Hambg. 79, 25-35 (2009) · Zbl 1191.20069 · doi:10.1007/s12188-008-0010-z
[23] Schmid, W.A.: Characterization of class groups of Krull monoids via their systems of sets of lengths: a status report. In: Adhikari, S.D., Ramakrishnan, B. (eds.) Number Theory and Applications. Proceedings of the International Conferences on Number Theory and Cryptography, pp. 189-212. Hindustan Book Agency, New Delhi (2009) · Zbl 1244.20053
[24] Schmid, W.A.: A realization theorem for sets of lengths. J. Number Theory 129, 990-999 (2009) · Zbl 1191.11031 · doi:10.1016/j.jnt.2008.10.019
[25] Smertnig, D.: Sets of lengths in maximal orders in central simple algebras. J. Algebr. 390, 1-43 (2013) · Zbl 1295.16023 · doi:10.1016/j.jalgebra.2013.05.016
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