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Original Articles

A characterization of finite abelian groups via sets of lengths in transfer Krull monoids

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Pages 4021-4041 | Received 04 Dec 2017, Published online: 26 Feb 2018

References

  • Baginski, P., Geroldinger, A., Grynkiewicz, D., Philipp, A. (2013). Products of two atoms in Krull monoids and arithmetical characterizations of class groups. Eur. J. Combin. 34:1244–1268. Special Issue in memory of Yahya Ould Hamidoune.
  • Chang, G. W. (2011). Every divisor class of Krull monoid domains contains a prime ideal. J. Algebra 336:370–377.
  • Fan, Y., Geroldinger, A., Kainrath, F., Tringali, S. (2017). Arithmetic of commutative semigroups with a focus on semigroups of ideals and modules. J. Algebra Appl. 16(11):42 p.
  • Fan, Y., Tringali, S. Power monoids: a bridge between factorization theory and arithmetic combinatorics. Arxiv:1701.09152.
  • Gao, W., Wang, L. (2012). On the maximal cross number of unique factorization zero-sum sequences over a finite abelian group. Integers 12, Paper A14:8 p.
  • Geroldinger, A. (2016). Sets of lengths. Am. Math. Monthly 123:960–988.
  • Geroldinger, A., Halter-Koch, F. (2006). Non-Unique Factorizations. Algebraic, Combinatorial and Analytic Theory. Pure and Applied Mathematics, Vol. 278. Boca Raton, FL: Chapman & Hall/CRC.
  • Geroldinger, A., Ruzsa, I. (2009). Combinatorial Number Theory and Additive Group Theory. Advanced Courses in Mathematics - CRM Barcelona. Basel, Switzerland: Birkhäuser.
  • Geroldinger, A., Schmid, W. A. A characterization of class groups via sets of lengths. ArXiv:1503.04679.
  • Geroldinger, A., Schmid, W. A., Zhong, Q. (2017). Systems of sets of lengths: transfer Krull monoids versus weakly Krull monoids. In: Fontana, M., Frisch, S., Glaz, S., Francesca Tartarone, F., Zanardo, P., eds. Rings, Polynomials, and Modules. Springer.
  • Geroldinger, A., Yuan, P. (2012). The set of distances in Krull monoids. Bull. London Math. Soc. 44:1203–1208.
  • Geroldinger, A., Zhong, Q. (2017). A characterization of class groups via sets of lengths II. J. Théor. Nombres Bordx. 29:327–346.
  • Geroldinger, A., Zhong, Q. (2018). Long sets of lengths with maximal elasticity. Canad. J. Math., to appear. DOI: 10.4153/CJM-2017-043-..
  • Geroldinger, A., Zhong, Q. (2016). The set of minimal distances in Krull monoids. Acta Arith. 173:97–120.
  • He, X. (2014). Cross number invariants of finite abelian groups. J. Number Theory 136:100–117.
  • Kim, B. (2015). The cross number of minimal zero-sum sequences over finite abelian groups. J. Number Theory 157:99–122.
  • Kim, H., Park, Y. S. (2001). Krull domains of generalized power series. J. Algebra 237:292–301.
  • Kriz, D. (2013). On a conjecture concerning the maximal cross number of unique factorization indexed sequences. J. Number Theory 133:3033–3056.
  • Krause, U., Zahlten, C. (1991). Arithmetic in Krull monoids and the cross number of divisor class groups. Mitt. Math. Ges. Hamb. 12:681–696.
  • Plagne, A., Schmid, W. A. (2005). On the maximal cardinality of half-factorial sets in cyclic groups. Math. Ann. 333:759–785.
  • Plagne, A., Schmid, W. A. (2018). On congruence half-factorial Krull monoids with cyclic class group, submitted.
  • Schmid, W. A. (2005). Differences in sets of lengths of Krull monoids with finite class group. J. Thé Nombres Bordx. 17:323–345.
  • Schmid, W. A. (2009). Arithmetical characterization of class groups of the form ℤ∕nℤ⊕ℤ∕nℤ via the system of sets of lengths. Abh. Math. Semin. Univ. Hamb. 79:25–35.
  • Schmid, W. A. (2009). Characterization of class groups of Krull monoids via their systems of sets of lengths : a status report, number Theory and applications. In: Adhikari, S. D., Ramakrishnan, B., eds. Proceedings of the International Conferences on Number Theory and Cryptography. New Delhi, India: Hindustan Book Agency, pp. 189–212.
  • Smertnig, D. (2013). Sets of lengths in maximal orders in central simple algebras. J. Algebra 390:1–43.
  • Smertnig, D. Factorizations in bounded hereditary noetherian prime rings. Proc. Edinburgh Math. Soc., to appear. ArXiv:1605.09274.
  • Zhong, Q. (2017). Sets of minimal distances and characterizations of class groups of Krull monoids. Ramanujan J., to appear. DOI:10.1007/s11139-016-9873-..