Elasticity of factorizations in atomic monoids and integral domains
Journal de Théorie des Nombres de Bordeaux, Volume 7 (1995) no. 2, pp. 367-385.

For an atomic domain R, its elasticity ρ(R) is defined by : ρ ( R ) = sup { m / n | u 1 u m = v 1 v n for irreducible u j , v i 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 u m = u 1 v n for irreducible u j , v i 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.

L'élasticité ρ(R) d'un anneau d'intégrité atomique R est définie par : ρ ( R ) = sup { m / n | u 1 u m = u 1 v n pour u i et v j irréductibles dans } . 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 u m = u 1 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.

@article{JTNB_1995__7_2_367_0,
     author = {Franz Halter-Koch},
     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},
     doi = {10.5802/jtnb.147},
     zbl = {0844.11068},
     mrnumber = {1378586},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.147/}
}
TY  - JOUR
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  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.147/
UR  - https://zbmath.org/?q=an%3A0844.11068
UR  - https://www.ams.org/mathscinet-getitem?mr=1378586
UR  - https://doi.org/10.5802/jtnb.147
DO  - 10.5802/jtnb.147
LA  - en
ID  - JTNB_1995__7_2_367_0
ER  - 
%0 Journal Article
%T Elasticity of factorizations in atomic monoids and integral domains
%J Journal de Théorie des Nombres de Bordeaux
%D 1995
%P 367-385
%V 7
%N 2
%I Université Bordeaux I
%U https://doi.org/10.5802/jtnb.147
%R 10.5802/jtnb.147
%G en
%F JTNB_1995__7_2_367_0
Franz Halter-Koch. Elasticity of factorizations in atomic monoids and integral domains. Journal de Théorie des Nombres de Bordeaux, Volume 7 (1995) no. 2, pp. 367-385. doi : 10.5802/jtnb.147. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.147/

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Cited by Sources: