Elasticity of factorizations in atomic monoids and integral domains

Halter-Koch, Franz

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

L'élasticité $\rho \left(R\right)$ d'un anneau d'intégrité atomique $R$ est définie par : $\rho \left(R\right)=sup\{m/n|u_1\cdots u_m=u_1\cdots v_n\text{pour}{u}_i\text{et}{v}_j\text{irréductibles}\text{dans}ℝ\}$. 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 ${\mu }_m\left(R\right)$ définis par : ${\mu }_m\left(R\right)=sup\left\{n\right|u_1\cdots u_m=u_1\cdots v_n\text{pour}{u}_i\text{et}{v}_j\text{irréductibles}\text{dans}R\}$. 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 ${\mu }_m$ et $\rho$ des monoïdes et des anneaux d'intégrité qui ont un intérêt propre.

For an atomic domain $R$, its elasticity $\rho \left(R\right)$ is defined by : $\rho \left(R\right)=sup\{m/n|u_1\cdots u_m=v_1\cdots v_n\text{for}\text{irreducible}{u}_j,v_i\in R\}$. 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 ${\mu }_m\left(R\right)$ defined by : ${\mu }_m\left(R\right)=sup\left\{n\right|u_1\cdots u_m=u_1\cdots v_n\text{for}\text{irreducible}{u}_j,v_i\in R\}$ . 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 ${\mu }_m$ and $\rho$ for monoids and integral domains which are of independent interest.

[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