N. R. Baeth and A. Geroldinger, Monoids of modules and arithmetic of direct-sum decompositions, Pacific Journal of Mathematics, vol.80, issue.2, pp.257-319, 2014.
DOI : 10.1016/0022-4049(94)90093-0

L. Carlitz, A characterization of algebraic number fields with class number two, Proc. Amer, pp.391-392, 1960.

S. Chang, S. T. Chapman, and W. W. Smith, Abstract, Mathematica Slovaca, vol.55, issue.2, pp.415-454, 2007.
DOI : 10.1201/9781420028249.ch22
URL : https://hal.archives-ouvertes.fr/hal-01256309

S. Chang, S. T. Chapman, and W. W. Smith, On minimum delta set values in block monoids over cyclic groups, The Ramanujan Journal, vol.55, issue.1, pp.155-171, 2007.
DOI : 10.1007/s11139-006-9000-x

S. T. Chapman, On the Davenport constant, the cross number, and their application in factorization theory, Zero-dimensional commutative rings, pp.167-190, 1994.

S. T. Chapman, U. Krause, and E. Oeljeklaus, On diophantine monoids and their class groups, Pacific Journal of Mathematics, vol.41, issue.1, pp.125-147, 2002.
DOI : 10.1007/978-1-4899-6752-7
URL : http://msp.org/pjm/2002/207-1/pjm-v207-n1-p06-s.pdf

S. T. Chapman and W. W. Smith, Factorization in dedekind domains with finite class group, Israel Journal of Mathematics, vol.82, issue.6, pp.65-95, 1990.
DOI : 10.1007/978-3-642-88405-4

S. T. Chapman and W. W. Smith, On a characterization of algebraic number fields with class number less than three, Journal of Algebra, vol.135, issue.2, pp.381-387, 1990.
DOI : 10.1016/0021-8693(90)90295-Y

J. Coykendall and W. W. Smith, On unique factorization domains, Journal of Algebra, vol.332, issue.1, pp.62-70, 2011.
DOI : 10.1016/j.jalgebra.2010.10.024
URL : http://doi.org/10.1016/j.jalgebra.2010.10.024

K. Cziszter, M. Domokos, and A. Geroldinger, The Interplay of Invariant Theory with??Multiplicative Ideal Theory and with Arithmetic Combinatorics, Multiplicative Ideal Theory and Factorization Theory, pp.43-95, 2016.
DOI : 10.1142/S1793042112500741

G. Freiman and A. Geroldinger, An Addition Theorem and Its Arithmetical Application, Journal of Number Theory, vol.85, issue.1, pp.59-73, 2000.
DOI : 10.1006/jnth.2000.2521
URL : http://doi.org/10.1006/jnth.2000.2521

W. Gao and A. Geroldinger, Systems of sets of lengths II, Abhandlungen aus dem Mathematischen Seminar der Universit??t Hamburg, vol.46, issue.1, pp.31-49505, 1988.
DOI : 10.4153/CMB-1975-004-4

A. Geroldinger, On nonunique factorizations into irreducible elements. II, Number theory, pp.723-757, 1987.

A. Geroldinger, Systeme von l??ngenmengen, Abhandlungen aus dem Mathematischen Seminar der Universit??t Hamburg, vol.46, issue.1, pp.115-130, 1990.
DOI : 10.1007/BF02941052

A. Geroldinger, Additive group theory and non-unique factorizations, Combinatorial Number Theory and Additive Group Theory, Advanced Courses in Mathematics. CRM Barcelona, pp.1-86, 2009.
DOI : 10.1201/9781420003208

A. Geroldinger and D. J. Grynkiewicz, On the arithmetic of Krull monoids with finite Davenport constant, Journal of Algebra, vol.321, issue.4, pp.1256-1284, 2009.
DOI : 10.1016/j.jalgebra.2008.11.020

A. Geroldinger and F. Halter-koch, Congruence monoids, Acta Arithmetica, vol.112, issue.3, pp.263-296, 2004.
DOI : 10.4064/aa112-3-4

A. Geroldinger and F. Halter-koch, Non-unique factorizations. Algebraic, Combinatorial and Analytic Theory, 2006.

A. Geroldinger and Y. O. Hamidoune, Zero-sumfree sequences in cyclic groups and some arithmetical application, Journal de Th??orie des Nombres de Bordeaux, vol.14, issue.1, pp.221-239, 2002.
DOI : 10.5802/jtnb.355
URL : http://www.numdam.org/article/JTNB_2002__14_1_221_0.pdf

A. Geroldinger and F. Kainrath, On the arithmetic of tame monoids with applications to Krull monoids and Mori domains, Journal of Pure and Applied Algebra, vol.214, issue.12, pp.2199-2218, 2010.
DOI : 10.1016/j.jpaa.2010.02.023

A. Geroldinger, F. Kainrath, and A. Reinhart, Arithmetic of seminormal weakly Krull monoids and domains, Journal of Algebra, vol.444, pp.201-245, 2015.
DOI : 10.1016/j.jalgebra.2015.07.026
URL : http://arxiv.org/abs/1508.00710

A. Geroldinger and W. A. Schmid, A characterization of class groups via sets of lengths II, Journal de Th??orie des Nombres de Bordeaux, vol.29, issue.2
DOI : 10.5802/jtnb.983

A. Geroldinger and Q. Zhong, The set of minimal distances in Krull monoids, Acta Arithmetica, vol.173, pp.97-120, 2016.
DOI : 10.4064/aa7906-1-2016

A. Geroldinger and Q. Zhong, A characterization of class groups via sets of lengths II, Journal de Th??orie des Nombres de Bordeaux, vol.29, issue.2, pp.327-346, 2017.
DOI : 10.5802/jtnb.983

R. Gilmer, W. Heinzer, and W. W. Smith, On the distribution of prime ideals within the ideal class group, Houston J. Math, vol.22, issue.1, pp.51-59, 1996.

F. Halter-koch, Ideal systems, volume 211 of Monographs and Textbooks in Pure and Applied Mathematics, 1998.

U. Krause, On monoids of finite real character, Proceedings of the American Mathematical Society, vol.105, issue.3, pp.546-554, 1989.
DOI : 10.1090/S0002-9939-1989-0953009-9

G. J. Leuschke and R. Wiegand, Cohen-Macaulay representations, volume 181 of Mathematical Surveys and Monographs

W. Narkiewicz, Finite abelian groups and factorization problems, Colloq. Math, vol.42, pp.319-330, 1979.

W. Narkiewicz, Elementary and analytic theory of algebraic numbers, 2004.
DOI : 10.1007/978-3-662-07001-7

A. Plagne and W. A. Schmid, On the maximal cardinality of half-factorial sets in cyclic groups, Mathematische Annalen, vol.37, issue.4, pp.759-785, 2005.
DOI : 10.1007/s00208-005-0690-y

M. Radziejewski and W. A. Schmid, Weakly half-factorial sets in finite abelian groups, Forum Mathematicum, vol.46, issue.4, pp.727-747, 2007.
DOI : 10.1090/S0002-9904-1976-14130-4

W. A. Schmid, Differences in sets of lengths of Krull monoids with finite class group, Journal de Th??orie des Nombres de Bordeaux, vol.17, issue.1, pp.323-345, 2005.
DOI : 10.5802/jtnb.493

W. A. Schmid, Periods of sets of lengths: a quantitative result and an associated inverse problem, Colloquium Mathematicum, vol.113, issue.1, pp.33-53, 2008.
DOI : 10.4064/cm113-1-4

W. A. Schmid, Arithmetical characterization of class groups of the form ???/n?????????/n??? via the system of sets of lengths, Abhandlungen aus dem Mathematischen Seminar der Universit??t Hamburg, vol.113, issue.1, pp.25-35, 2009.
DOI : 10.5802/jtnb.493

W. A. Schmid, A realization theorem for sets of lengths, Journal of Number Theory, vol.129, issue.5, pp.990-999, 2009.
DOI : 10.1016/j.jnt.2008.10.019

W. A. Schmid, HIGHER-ORDER CLASS GROUPS AND BLOCK MONOIDS OF KRULL MONOIDS WITH TORSION CLASS GROUP, Journal of Algebra and Its Applications, vol.54, issue.03, pp.433-464, 2010.
DOI : 10.1007/BFb0090545

L. Skula, On c-semigroups [40] J. ´ Sliwa. Factorizations of distinct lengths in algebraic number fields, 41] J. ´ Sliwa. Remarks on factorizations in algebraic number fields, pp.247-257399, 1976.

D. Smertnig, Sets of lengths in maximal orders in central simple algebras, Journal of Algebra, vol.390, pp.1-43, 2013.
DOI : 10.1016/j.jalgebra.2013.05.016

A. Zaks, Half factorial domains, Bulletin of the American Mathematical Society, vol.82, issue.5, pp.721-723, 1976.
DOI : 10.1090/S0002-9904-1976-14130-4

Q. Zhong, Sets of minimal distances and characterizations of class groups of Krull monoids, The Ramanujan Journal, vol.390
DOI : 10.1016/j.jalgebra.2013.05.016