Long Sets of Lengths With Maximal Elasticity

Geroldinger, Alfred ; Zhong, Qinghai

Canadian journal of mathematics, Tome 70 (2018) no. 6, pp. 1284-1318

Résumé

We introduce a newinvariant describing the structure of sets of lengths in atomicmonoids and domains. For an atomic monoid $H$ , let ${{\Delta }_{\rho }}\left( H \right)$ be the set of all positive integers d that occur as differences of arbitrarily long arithmetical progressions contained in sets of lengths havingmaximal elasticity $\rho \left( H \right)$ . We study ${{\Delta }_{\rho }}\left( H \right)$ for transfer Krull monoids of finite type (including commutative Krull domains with finite class group) with methods from additive combinatorics, and also for a class of weakly Krull domains (including orders in algebraic number fields) for which we use ideal theoretic methods.

Détail

DOI : 10.4153/CJM-2017-043-4

Mots-clés : 13A05, 13F05, 16H10, 16U30, 20M13, transfer Krull monoid, weakly Krull monoid, set of length, elasticity

Bibliographie

[1] [1] Anderson, D. D., Anderson, D. E., and Zafrullah, M., Atomic domains in which almost all atoms are prime. Comm. Algebra 20 (1992), 1447–1462. http://dx.doi.Org/10.1080/00927879208824413 Google Scholar

[2] [2] Anderson, D. D., Mott, J., and Zafrullah, M., Finite character representations for integral domains. Boll. Un. Mat. Ital. B. 6 (1992), 613–630. Google Scholar

[3] [3] Baeth, N. R. and Smertnig, D., Factorization theory: from commutative to noncommutative settings. J. Algebra 441 (2015), 475–551. http://dx.doi.Org/10.1016/j.jalgebra.2O15.06.007 Google Scholar

[4] [4] Barucci, V., D'Anna, M., and Frôberg, R., Analytically unramified one-dimensional semilocal rings and their value semigroups. J. Pure Appl. Algebra 147 (2000), 215–254. http://dx.doi.Org/10.1016/S0022-4049(98)00160-1 Google Scholar

[5] [5] Barucci, V., Dobbs, D. E., and Fontana, M., Maximality properties in numerical semigroups and applications to one-dimensional analytically irreducible local domains. Mem. Amer. Math. Soc, 125. 1997. Google Scholar

[6] [6] Bowles, C., Chapman, S. T., Kaplan, N., and Reiser, D., On delta sets of numerical monoids. J. Algebra Appl. 5 (2006), 695–718. http://dx.doi.Org/10.1142/S0219498806001958 Google Scholar

[7] [7] Chang, S., Chapman, S. T., and Smith, W. W., On minimum delta set values in block monoids over cyclic groups. Ramanujan J. 14 (2007), 155–171. http://dx.doi.Org/10.1007/s11139-006-9000-x Google Scholar

[8] [8] Chapman, S. T., Fontana, M., Geroldinger, A., and Olberding, B. (eds.), Multiplicative ideal theory and factorization theory. Springer Proceedings in Mathematics and Statistics, 170. Springer, 2016. Google Scholar

[9] [9] Chapman, S. T., Holden, M., and Moore, T., Full elasticity in atomic monoids and integral domains. Rocky Mountain J. Math. 36 (2006), 1437–1455. http://dx.doi.Org/10.1216/rmjm/1181069375 Google Scholar

[10] [10] D'Anna, M., Ring and semigroup constructions. In: Multiplicative ideal theory and factorization theory. Springer Proc. Math. Stat, 170, Springer, 2016, pp. 97–115. http://dx.doi.Org/10.1007/978-3-319-38855-7_4 Google Scholar

[11] [11] Fan, Y. and Tringali, S., Power monoids: a bridge between factorization theory and arithmetic combinatorics. arxiv:1701.09152. Google Scholar

[12] [12] Fan, Y. and Zhong, Q., Products ofk atoms in Krull monoids. Monatsh. Math. 181 (2016), 779–795. http://dx.doi.org/10.1007/s00605-016-0942-9 Google Scholar

[13] [13] Fontana, M., Houston, E., and Lucas, T., Factoring ideals in integral domains. Lecture Notes of the Unione Matematica Italiana, 14. Springer, Heidelberg, 2013. Google Scholar

[14] [14] Geroldinger, A., On non-unique factorizations into irreducible elements. II. Colloq.Math. Soc. Janos Bolyai, 51. North Holland, Amsterdam, 1990, pp. 723-757. Google Scholar

[15] [15] Geroldinger, A., Sets of lengths. Amer. Math. Monthly 123 (2016), 960–988. Google Scholar

[16] [16] Geroldinger, A., Grynkiewicz, D. J., and Yuan, P., On products ofk atoms II. Mosc. J. Comb. Number Theory 5 (2015), 73–129. Google Scholar

[17] [17] Geroldinger, A. and Halter-Koch, F., Non-unique factorizations, algebraic, combinatorial and analytic theory. Pure and Applied Mathematics, 278 Chapman and Hall/CRC, Boca Raton, FL, 2006. Google Scholar

[18] [18] Geroldinger, A. and Kainrath, F., On the arithmetic of tame monoids with applications to Krull monoids and Mori domains. J. Pure Appl. Algebra 214 (2010), 2199–2218. http://dx.doi.org/!0.1016/j.jpaa.2O10.02.023 Google Scholar

[19] [19] Geroldinger, A., Kainrath, E., and Reinhart, A., Arithmetic of seminormal weakly Krull monoids and domains. J. Algebra 444 (2015), 201–245. http://dx.doi.Org/10.1016/j.jalgebra.2O15.07.026 Google Scholar

[20] [20] Geroldinger, A. and Schmid, W. A., A characterization of class groups via sets of lengths. arxiv:/1 503.04679 Google Scholar

[21] [21] Geroldinger, A., Schmid, W. A., and Zhong, Q., Systems of sets of lengths: transfer Krull monoids versus weakly Krull monoids. In: Rings, polynomials, and modules. Springer, 2017. arxiv:1 606.05063 Google Scholar

[22] [22] Geroldinger, A. and Yuan, P., The set of distances in Krull monoids. Bull. Lond. Math. Soc. 44 (2012), 1203–1208. http://dx.doi.Org/10.1112/blms/bds046 Google Scholar

[23] [23] Geroldinger, A. and Zhong, Q., The set of distances in seminormal weakly Krull monoids. J. Pure Appl. Algebra 220 (2016), 3713–3732. http://dx.doi.Org/10.1016/j.jpaa.2O16.05.009 Google Scholar

[24] [24] Geroldinger, A. and Zhong, Q., The set of minimal distances in Krull monoids. Acta Arith. 173 (2016), 97–120. Google Scholar

[25] [25] Geroldinger, A. and Zhong, Q., A characterization of class groups via sets of lengths II. J. Théor. Nombres Bordeaux. 29 (2017), 327–346. Google Scholar

[26] [26] Halter-Koch, F., Divisor theories with primary elements and weakly Krull domains. Boll. Un. Mat. Ital. B 9 (1995), 417–441. Google Scholar

[27] [27] Halter-Koch, F., Elasticity of factorizations in atomic monoids and integral domains. J. Théor. Nombres Bordeaux. 7 (1995), 367–385. http://dx.doi.Org/10.58O2/jtnb.147 Google Scholar

[28] [28] Halter-Koch, F., Ideal systems. An introduction to multiplicative ideal theory. Monographs and Textbooks in Pure and Applied Mathematics, 211. Marcel Dekker, New York, 1998. Google Scholar

[29] [29] Halter-Koch, F., Quadratic Irrationals. Pure and Applied Mathematics, 306. Chapman and Hall/CRC, Boca Ration, FL, 2013. Google Scholar

[30] [30] Kainrath, F., A note on quotients formed by unit groups of semilocal rings. Houston J. Math. 24 (1998), 613–618. Google Scholar

[31] [31] Kainrath, F., Arithmetic of Mori domains and monoids: the global case. In: Multiplicative ideal theory and factorization theory. Springer Proc. Math. Stat., 170. Springer, 2016, pp. 183–218. Google Scholar

[32] [32] Plagne, A. and Schmid, W. A., On congruence half-factorial Krull monoids with cyclic class group. arxiv:1 709.00859 Google Scholar

[33] [33] Schmid, W. A., The inverse problem associated to the Davenport constant for C ⊕ C ⊕ C, and applications to the arithmetical characterization of class groups. Electron. J. Comb. 18 (2011), Research Paper 33. Google Scholar

[34] [34] Schmid, W. A., Some recent results and open problems on sets of lengths of Krull monoids with finite class group. In: Multiplicative ideal theory and factorization theory. Springer Proc. Math. Stat., 170. Springer, 2016, pp. 323–352. Google Scholar

[35] [35] Smertnig, D., Factorizations in bounded hereditary noetherianprime rings. arxiv:1605.09274 Google Scholar

[36] [36] Smertnig, D., Sets of lengths in maximal orders in central simple algebras. J. Algebra 390 (2013), 1–43. http://dx.doi.Org/10.1016/j.jalgebra.2013.05.016 Google Scholar

[37] [37] Zhong, Q., Sets of minimal distances and characterizations of class groups of Krull monoids. Ramanujan J., to appear. Google Scholar