Elasticity of factorizations in atomic monoids and integral domains

Halter-Koch, Franz

Halter-Koch, Franz

Journal de Théorie des Nombres de Bordeaux, Tome 7 (1995) no. 2, pp. 367-385.

Résumé
Abstract

L'élasticité $ρ (R)$ d'un anneau d'intégrité atomique $R$ est définie par : $ρ (R) = sup {m / n | u_{1} \dots u_{m} = u_{1} \dots v_{n} pour u_{i} et v_{j} irréductibles dans ℝ}$ . Nous étudions ici l'élasticité des anneaux d'intégrité noethériens au moyen des invariants plus fins. Nous étudions ici l'élasticité des anneaux d'intégrité noethériens au moyen des invariants plus fins $μ_{m} (R)$ définis par : $μ_{m} (R) = sup {n | u_{1} \dots u_{m} = u_{1} \dots v_{n} pour u_{i} et v_{j} irréductibles dans R}$ . Le résultat principal que nous donnons permet de caractériser les anneaux d'entiers des corps de nombres qui ont une élasticité finie. Chemin faisant nous obtenons une série de résultats sur les invariants $μ_{m}$ et $ρ$ des monoïdes et des anneaux d'intégrité qui ont un intérêt propre.

For an atomic domain $R$ , its elasticity $ρ (R)$ is defined by : $ρ (R) = sup {m / n | u_{1} \dots u_{m} = v_{1} \dots v_{n} for irreducible u_{j}, v_{i} \in R}$ . We study the elasticity of one-dimensional noetherian domains by means of the more subtle invariants. We study the elasticity of one-dimensional noetherian domains by means of the more subtle invariants $μ_{m} (R)$ defined by : $μ_{m} (R) = sup {n | u_{1} \dots u_{m} = u_{1} \dots v_{n} for irreducible u_{j}, v_{i} \in R}$ . As a main result we characterize all orders in algebraic number fields having finite elasticity. On the way, we obtain a series of results concerning the invariants $μ_{m}$ and $ρ$ for monoids and integral domains which are of independent interest.

MR 1378586 | Zbl 0844.11068 | 2 citations dans Numdam

BibTeX
RIS

@article{JTNB_1995__7_2_367_0,
     author = {Halter-Koch, Franz},
     title = {Elasticity of factorizations in atomic monoids and integral domains},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {367--385},
     publisher = {Universit\'e Bordeaux I},
     volume = {7},
     number = {2},
     year = {1995},
     zbl = {0844.11068},
     mrnumber = {1378586},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_1995__7_2_367_0/}
}

TY  - JOUR
AU  - Halter-Koch, Franz
TI  - Elasticity of factorizations in atomic monoids and integral domains
JO  - Journal de Théorie des Nombres de Bordeaux
PY  - 1995
DA  - 1995///
SP  - 367
EP  - 385
VL  - 7
IS  - 2
PB  - Université Bordeaux I
UR  - http://www.numdam.org/item/JTNB_1995__7_2_367_0/
UR  - https://zbmath.org/?q=an%3A0844.11068
UR  - https://www.ams.org/mathscinet-getitem?mr=1378586
LA  - en
ID  - JTNB_1995__7_2_367_0
ER  -

Halter-Koch, Franz. Elasticity of factorizations in atomic monoids and integral domains. Journal de Théorie des Nombres de Bordeaux, Tome 7 (1995) no. 2, pp. 367-385. http://www.numdam.org/item/JTNB_1995__7_2_367_0/

Bibliographie
Cité par

[1] D.D. Anderson And D.F. Anderson, Elasticity of factorizations in integral domains, J. Pure Appl. Algebra 80 (1992), 217-235. | MR 1170712 | Zbl 0773.13003

[2] D.D. Anderson And D.F. Anderson, Elasticity of factorizations in irategral domains II, Houston J. Math. 20 (1994), 1-15. | MR 1272557 | Zbl 0838.13002

[3] D.D. Anderson, D.F. Anderson, S.T. Chapman and W.W. Smith, Rational Elasticity of Factorizations in Krull Domains, Proc. AMS 117 (1993), 37 -43. | MR 1106176 | Zbl 0777.13003

[4] D.D. Anderson and J.L. Mott, Cohen-Kaplansky Domains: Integral Domains with a Finite Number of Irreducible Elements, J. Algebra 148 (1992), 17-41. | MR 1161563 | Zbl 0773.13005

[5] D D. Anderson, J.L. Mott and M. Zafrullah, Finite character representations for integral domains, Boll. U. M. I. 6-B (1992), 613- 630. | MR 1191956 | Zbl 0773.13004

[6] A. Fröhlich, Local fields, Algebraic Number Theory (J. W. S. Cassels and A. Fröhlich, eds.), Academic Press 1967. | MR 236145

[7] A. Geroldinger, On the arithmetic of certain not integrally closed noetherian domains, Comm. Algebra 19 (1991), 685-698. | MR 1100372 | Zbl 0723.20038

[8] A. Geroldinger, T-block monoids and their arithmetical applications to noetherian domains, Comm. Algebra 22 (1994), 1603-1615. | MR 1264730 | Zbl 0809.13013

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

[10] A. Geroldinger and F. Halter-Koch, Realization Theorems for Semigroups with Divisor Theory, Semigroup Forum 44 (1992), 229-237. | MR 1141841 | Zbl 0751.20045

[11] A. Geroldinger and F. Halter-Koch, Arithmetical theory of monoid hommnorphisms, Semigroup Forum 48 (1994), 333-362. | MR 1262998 | Zbl 0805.20057

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

[13] A. Geroldinger and R. Schneider, On Davenport's constant, J. Comb. Theory, Series A 61 (1992), 147-152. | MR 1178393 | Zbl 0759.20008

[14] F. Halter-Koch, Ein Approximationssatz für Halbgruppen mit Divisorentheorie, Result. Math 19 (1991), 74-82. | MR 1091957 | Zbl 0742.20060

[15] F. Halter-Koch, Divisor theories with primary elements and weakly Krull domains, Boll. U. M. I.. | Zbl 0849.20041

[16] F. Halter-Koch, Zur Zahlen- und Idealtheorie eindimensionaler noetherscher Integritätsbereiche, J. Algebra 136 (1991), 103-108. | MR 1085124 | Zbl 0743.13001

[17] N. Jacobson, Basic Algebra I, Freeman and Co,1974. | MR 356989 | Zbl 0284.16001

[18] J. Neukirch, Algebraische Zahlentheorie, Springer, 1992. | Zbl 0747.11001

[19] J.L. Steffan, Longeurs des décompositions en produits d'éléments irréducibles dans un anneau de Dedekind, J. Algebra 102 (1986), 229-236. | MR 853241 | Zbl 0593.13015

[20] R.J. Valenza, Elasticity of factorizations in number fields, J. Number Theory 39 (1990), 212-218. | MR 1072466 | Zbl 0721.11043