Differences in sets of lengths of Krull monoids with finite class group
Journal de Théorie des Nombres de Bordeaux, Volume 17 (2005) no. 1, pp. 323-345.

Let H be a Krull monoid with finite class group where every class contains some prime divisor. It is known that every set of lengths is an almost arithmetical multiprogression. We investigate which integers occur as differences of these progressions. In particular, we obtain upper bounds for the size of these differences. Then, we apply these results to show that, apart from one known exception, two elementary p-groups have the same system of sets of lengths if and only if they are isomorphic.

Soit H un monoïde de Krull dont le groupe de classes est fini. On suppose que chaque classe contient un diviseur premier. On sait que tout ensemble de longueurs est une presque multiprogression arithmétique. Nous étudions les nombres entiers qui apparaissent comme raison de ces progressions. Nous obtenons en particulier une borne supérieure sur la taille de ces raisons. En appliquant ces résultats, nous pouvons montrer que, sauf dans un cas particulier connu, deux p-groupes élémentaires ont le même système d’ensembles de longueurs si et seulement si ils sont isomorphes.

Published online:
DOI: 10.5802/jtnb.493
Wolfgang A. Schmid 1

1 Institut für Mathematik und Wissenschaftliches Rechnen Karl-Franzens-Universität Graz Heinrichstraße 36 8010 Graz, Austria
@article{JTNB_2005__17_1_323_0,
     author = {Wolfgang A. Schmid},
     title = {Differences in sets of lengths of {Krull} monoids with finite class group},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {323--345},
     publisher = {Universit\'e Bordeaux 1},
     volume = {17},
     number = {1},
     year = {2005},
     doi = {10.5802/jtnb.493},
     zbl = {1090.20034},
     mrnumber = {2152227},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.493/}
}
TY  - JOUR
TI  - Differences in sets of lengths of Krull monoids with finite class group
JO  - Journal de Théorie des Nombres de Bordeaux
PY  - 2005
DA  - 2005///
SP  - 323
EP  - 345
VL  - 17
IS  - 1
PB  - Université Bordeaux 1
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.493/
UR  - https://zbmath.org/?q=an%3A1090.20034
UR  - https://www.ams.org/mathscinet-getitem?mr=2152227
UR  - https://doi.org/10.5802/jtnb.493
DO  - 10.5802/jtnb.493
LA  - en
ID  - JTNB_2005__17_1_323_0
ER  - 
%0 Journal Article
%T Differences in sets of lengths of Krull monoids with finite class group
%J Journal de Théorie des Nombres de Bordeaux
%D 2005
%P 323-345
%V 17
%N 1
%I Université Bordeaux 1
%U https://doi.org/10.5802/jtnb.493
%R 10.5802/jtnb.493
%G en
%F JTNB_2005__17_1_323_0
Wolfgang A. Schmid. Differences in sets of lengths of Krull monoids with finite class group. Journal de Théorie des Nombres de Bordeaux, Volume 17 (2005) no. 1, pp. 323-345. doi : 10.5802/jtnb.493. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.493/

[1] D.D. Anderson, (editor), Factorization in integral domains. Lecture Notes in Pure and Applied Mathematics 189, Marcel Dekker Inc., New York, 1997. | MR: 1460766 | Zbl: 0865.00039

[2] D.F. Anderson, Elasticity of factorizations in integral domains: a survey. In [1], 1–29. | MR: 1460767 | Zbl: 0773.13003

[3] L. Carlitz, A characterization of algebraic number fields with class number two. Proc. Amer. Math. Soc. 11 (1960), 391–392. | MR: 111741 | Zbl: 0202.33101

[4] S. Chapman, A. Geroldinger, Krull domains and monoids, their sets of lengths and associated combinatorial problems. In [1], 73–112. | MR: 1460769 | Zbl: 0897.13001

[5] P. van Emde Boas, A combinatorial problem on finite abelian groups II. Report ZW-1969-007, Math. Centre, Amsterdam (1969), 60p. | MR: 255672 | Zbl: 0203.32703

[6] W. Gao, A. Geroldinger, Half-factorial domains and half-factorial subsets in abelian groups. Houston J. Math. 24 (1998), 593–611. | MR: 1686626 | Zbl: 0994.20046

[7] W. Gao, A. Geroldinger, Systems of sets of lengths II. Abh. Math. Sem. Univ. Hamburg 70 (2000), 31–49. | MR: 1809532 | Zbl: 1036.11054

[8] A. Geroldinger, Über nicht-eindeutige Zerlegungen in irreduzible Elemente. Math. Z. 197 (1988), 505–529. | MR: 932683 | Zbl: 0618.12002

[9] A. Geroldinger, Systeme von Längenmengen. Abh. Math. Sem. Univ. Hamburg 60 (1990), 115–130. | MR: 1087122 | Zbl: 0721.11042

[10] A. Geroldinger, On nonunique factorizations into irreducible elements. II. Colloq. Math. Soc. János Bolyai 51, North-Holland, Amsterdam, 1990, 723–757. | MR: 1058242 | Zbl: 0703.11057

[11] A. Geroldinger, The cross number of finite abelian groups. J. Number Theory 48 (1994), 219–223. | MR: 1285540 | Zbl: 0814.20033

[12] A. Geroldinger, A structure theorem for sets of lengths. Colloq. Math. 78 (1998), 225–259. | MR: 1659136 | Zbl: 0926.11082

[13] A. Geroldinger, Y. ould Hamidoune, Zero-sumfree sequences in cyclic groups and some arithmetical application. Journal Théor. Nombres Bordeaux 14 (2002), 221–239. | Numdam | MR: 1925999 | Zbl: 1018.11011

[14] A. Geroldinger, G. Lettl, Factorization problems in semigroups. Semigroup Forum 40 (1990), 23–38. | MR: 1014223 | Zbl: 0693.20063

[15] A. Geroldinger, R. Schneider, The cross number of finite abelian groups III. Discrete Math. 150 (1996), 123–130. | MR: 1392725 | Zbl: 0848.20048

[16] F. Halter-Koch, Elasticity of factorizations in atomic monoids and integral domains. J. Théor. Nombres Bordeaux 7 (1995), 367–385. | Numdam | MR: 1378586 | Zbl: 0844.11068

[17] F. Halter-Koch, Finitely generated monoids, finitely primary monoids and factorization properties of integral domains. In [1], 31–72. | MR: 1460768 | Zbl: 0882.13027

[18] F. Halter-Koch, Ideal Systems. Marcel Dekker Inc., New York, 1998. | MR: 1828371 | Zbl: 0953.13001

[19] U. Krause, A characterization of algebraic number fields with cyclic class group of prime power order. Math. Z. 186 (1984), 143–148. | MR: 741299 | Zbl: 0522.12006

[20] W. Narkiewicz, Elementary and analytic theory of algebraic numbers, second edition. Springer-Verlag, Berlin, 1990. | MR: 1055830 | Zbl: 02107000

[21] J.E. Olson, A combinatorial problem on finite abelian groups, I,. J. Number Theory 1 (1969), 8–10. | MR: 237641 | Zbl: 0169.02003

[22] W.A. Schmid, Arithmetic of block monoids. Math. Slovaca 54 (2004), 503–526. | MR: 2114621 | Zbl: 02185567

[23] W.A. Schmid, Half-factorial sets in elementary p-groups. Far East J. Math. Sci. (FJMS), to appear. | MR: 2255073 | Zbl: 05073469

[24] L. Skula, On c-semigroups. Acta Arith. 31 (1976), 247–257. | MR: 444817 | Zbl: 0303.13014

[25] J. Śliwa, Factorizations of distinct length in algebraic number fields. Acta Arith. 31 (1976), 399–417. | MR: 429830 | Zbl: 0347.12005

[26] J. Śliwa, Remarks on factorizations in algebraic number fields. Colloq. Math. 46 (1982), 123–130. | MR: 672372 | Zbl: 0514.12005

[27] A. Zaks, Half factorial domains. Bull. Amer. Math. Soc. 82 (1976), 721–723. | MR: 407001 | Zbl: 0338.13020

Cited by Sources: