Let be a Krull monoid with finite class group and suppose that every class contains a prime divisor. If an element has a factorization into irreducible elements , then is called the length of the factorization and the set of all possible factorization lengths is the set of lengths of . It is classical that the system of all sets of lengths depends only on the class group , and a standing conjecture states that conversely the system is characteristic for the class group. We verify the conjecture if the class group is isomorphic to with and . Indeed, let be a further Krull monoid with class group such that every class contains a prime divisor and suppose that . We prove that, if one of the groups and is isomorphic to with as above, then and are isomorphic (apart from two well-known pairings).
Soit un monoïde de Krull avec un groupe des classes fini , et supposons que chaque classe contient un diviseur premier. Si un élément a une factorisation en éléments irréductibles , alors nous appelons la longueur de la factorisation et l’ensemble de toutes les longueurs de factorisation possibles l’ensemble des longueurs de . C’est bien connu que le système de tous les ensembles de longueurs ne dépend que du groupe des classes , et c’est bien une conjecture de longue date que, inversement, le système caractérise le groupe des classes. Nous vérifions la conjecture si le groupe des classes est isomorphe à avec et .
En effet, soit un autre monoïde de Krull avec un groupe des classes tel que chaque classe contient un diviseur premier, et supposons que . Nous montrons que, si l’un des groupes et est isomorphe à avec donnés comme ci-dessus, alors et sont isomorphes (à part deux exceptions bien connues).
Accepted:
Published online:
Mots-clés : Krull monoids, maximal orders, seminormal orders, class groups, arithmetical characterizations, sets of lengths, zero-sum sequences, Davenport constant
Alfred Geroldinger 1; Qinghai Zhong 1

@article{JTNB_2017__29_2_327_0, author = {Alfred Geroldinger and Qinghai Zhong}, title = {A characterization of class groups via sets of lengths {II}}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {327--346}, publisher = {Soci\'et\'e Arithm\'etique de Bordeaux}, volume = {29}, number = {2}, year = {2017}, doi = {10.5802/jtnb.983}, language = {en}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.983/} }
TY - JOUR AU - Alfred Geroldinger AU - Qinghai Zhong TI - A characterization of class groups via sets of lengths II JO - Journal de théorie des nombres de Bordeaux PY - 2017 SP - 327 EP - 346 VL - 29 IS - 2 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.983/ DO - 10.5802/jtnb.983 LA - en ID - JTNB_2017__29_2_327_0 ER -
%0 Journal Article %A Alfred Geroldinger %A Qinghai Zhong %T A characterization of class groups via sets of lengths II %J Journal de théorie des nombres de Bordeaux %D 2017 %P 327-346 %V 29 %N 2 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.983/ %R 10.5802/jtnb.983 %G en %F JTNB_2017__29_2_327_0
Alfred Geroldinger; Qinghai Zhong. A characterization of class groups via sets of lengths II. Journal de théorie des nombres de Bordeaux, Volume 29 (2017) no. 2, pp. 327-346. doi : 10.5802/jtnb.983. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.983/
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