본문 바로가기
KERD
논문KCI 등재

A characterization of class groups via sets of lengths

Alfred Geroldinger|Wolfgang Alexander Schmid
대한수학회 (The Korean Mathematical Society)|2019.07|대한수학회지 (Journal of the KMS)|제56권 제4호|pp. 869-915|DOI 10.4134/JKMS.j180467

2

조회

0

다운로드

0

인용

0

참고문헌

키워드

sets of lengthsKrull monoidsclass groupsarithmetical characterizationszero-sum sequences
주제분류수학

초록

Let $H$ be a Krull monoid with class group $G$ such that every class contains a prime divisor. Then every nonunit $a \in H$ can be written as a finite product of irreducible elements. If $a=u_1 \cdot\ldots\cdot u_k$ with irreducibles $u_1, \ldots, u_k \in H$, then $k$ is called the length of the factorization and the set $\mathsf L (a)$ of all possible $k$ is the set of lengths of $a$. It is well-known that the system $\mathcal L (H) = \{\mathsf L (a) \mid a \in H \}$ depends only on the class group $G$. We study the inverse question asking whether the system $\mathcal L (H)$ is characteristic for the class group. Let $H'$ be a further Krull monoid with class group $G'$ such that every class contains a prime divisor and suppose that $\mathcal L (H) = \mathcal L (H')$. We show that, if one of the groups $G$ and $G'$ is finite and has rank at most two, then $G$ and $G'$ are isomorphic (apart from two well-known exceptions).

Abstract

Let $H$ be a Krull monoid with class group $G$ such that every class contains a prime divisor. Then every nonunit $a \in H$ can be written as a finite product of irreducible elements. If $a=u_1 \cdot\ldots\cdot u_k$ with irreducibles $u_1, \ldots, u_k \in H$, then $k$ is called the length of the factorization and the set $\mathsf L (a)$ of all possible $k$ is the set of lengths of $a$. It is well-known that the system $\mathcal L (H) = \{\mathsf L (a) \mid a \in H \}$ depends only on the class group $G$. We study the inverse question asking whether the system $\mathcal L (H)$ is characteristic for the class group. Let $H'$ be a further Krull monoid with class group $G'$ such that every class contains a prime divisor and suppose that $\mathcal L (H) = \mathcal L (H')$. We show that, if one of the groups $G$ and $G'$ is finite and has rank at most two, then $G$ and $G'$ are isomorphic (apart from two well-known exceptions).

참고문헌 (38)

  1. P. C. Baayen, "(C2 ⊕ C2 ⊕ C2 ⊕ C2n)!", Math. Centrum Amsterdam Afd. Zuivere Wisk, 1969, pp.21~ (1969)
  2. N. R. Baeth, "Monoids of modules and arithmetic of direct-sum decompositions", Pacific J. Math, 271(2), pp.257~319 (2014)
  3. N. Baeth, "Atoms of the relative block monoid", Involve, 2(1), pp.29~36 (2009)
  4. N. R. Baeth, "Factorization theory and decompositions of modules", Amer. Math. Monthly, 120(1), pp.3~34 (2013)
  5. P. Baginski, "Products of two atoms in Krull monoids and arithmetical characterizations of class groups", European J. Combin, 34(8), pp.1244~1268 (2013)
  6. S. T. Chapman, "Multiplicative Ideal Theory and Factorization Theory" (2016), Springer
  7. W. Gao, "Systems of sets of lengths. II", Abh. Math. Sem. Univ. Hamburg, 70, pp.31~49 (2000)
  8. W. Gao, "On zero-sum sequences in Z", nZ ⊕ Z, Integers, pp.2003, 3(A8)[DOI]
  9. W. Gao, "Inverse zero-sum problems. III", Acta Arith, 141(2), pp.103~152 (2010)
  10. A. Geroldinger, "Systeme von Langenmengen", Abh. Math. Sem. Univ. Hamburg, 60, pp.115~130 (1990)
  11. A. Geroldinger, "Combinatorial number theory and additive group theory", pp.1~86 (2009), Birkhauser Verlag
  12. A. Geroldinger, "Sets of lengths", Amer. Math. Monthly, 123(10), pp.960~988 (2016)
  13. A. Geroldinger, "The catenary degree of Krull monoids I", J. Th´eor. Nombres Bordeaux, 23(1), pp.137~169 (2011)
  14. A. Geroldinger, "On products of k atoms II", Mosc. J. Comb. Number Theory, 5(3), pp.3~59 (2015)
  15. A. Geroldinger, "Non-unique Factorizations", pp.2006, Chapman & Hall
  16. Alfred Geroldinger, "ON v-MAROT MORI RINGS AND C-RINGS", 대한수학회지, 52(1), pp.1~21 (2015), 대한수학회
  17. A. Geroldinger, "Combinatorial number theory and additive group theory, Advanced Courses in Mathematics" (2009), Birkhauser Verlag
  18. A. Geroldinger, "The set of distances in Krull monoids", Bull. Lond. Math. Soc, 44(6), pp.1203~1208 (2012)
  19. A. Geroldinger, "The catenary degree of Krull monoids II", J. Aust. Math. Soc, 98(3), pp.324~354 (2015)
  20. A. Geroldinger, "The set of minimal distances in Krull monoids", Acta Arith, 173(2), pp.97~120 (2016)
  21. A. Geroldinger, "A characterization of class groups via sets of lengths II", J. Th´eor. Nombres Bordeaux, 29(2), pp.327~346 (2017)
  22. D. J. Grynkiewicz, "Structural Additive Theory" (2013), Springer
  23. F. Halter-Koch, "Factorization of algebraic integers", Grazer Math. Berichte, 191 (1983)
  24. F. Halter-Koch, "Relative block semigroups and their arithmetical applications", Comment. Math. Univ. Carolin, 33(3), pp.373~381 (1992)
  25. F. Halter-Koch, "Ideal Systems" (1998), Marcel Dekker, Inc
  26. J. Kaczorowski, "A pure arithmetical characterization for certain fields with a given class group", Colloq. Math, 45(2), pp.327~330 (1981)
  27. W. Narkiewicz, "Elementary and Analytic Theory of Algebraic Numbers" (1974), PWN—Polish Scientific Publishers
  28. C. Reiher, "A proof of the theorem according to which every prime number possesses property B", PhD, pp.2010
  29. D. E. Rush, "An arithmetic characterization of algebraic number fields with a given class group", Math. Proc. Cambridge Philos. Soc, 94(1), pp.23~28 (1983)
  30. W. A. Schmid, "Arithmetic of block monoids", Math. Slovaca, 54(5), pp.503~526 (2004)
  31. W. A. Schmid, "Periods of sets of lengths : a quantitative result and an associated inverse problem", Colloq. Math, 113(1), pp.33~53 (2008)
  32. W. A. Schmid, "Arithmetical characterization of class groups of the form Z", nZ ⊕Z, Abh. Math. Semin. Univ. Hambg, pp.2009, 79(1)[DOI]
  33. W. A. Schmid, "Number theory and applications", pp.189~212 (2009), Hindustan Book Agency
  34. W. A. Schmid, "A realization theorem for sets of lengths", J. Number Theory, 129(5), pp.990~999 (2009)
  35. W. A. Schmid, "Inverse zero-sum problems II", Acta Arith, 143(4), pp.333~343 (2010)
  36. W. A. Schmid, "The inverse problem associated to the Davenport constant for C2 ⊕ C2 ⊕ C2n, and applications t o the arithmetical characterization of class groups", Electron. J. Combin, 18(1), pp.42~ (2011)[DOI]
  37. Q. Zhong, "A characterization of finite abelian groups via sets of lengths in transfer Krull monoids", Comm. Algebra, 46(9), pp.4021~4041 (2018)
  38. Q. Zhong, "Sets of minimal distances and characterizations of class groups of Krull monoids", Ramanujan J, 45(3), pp.719~737 (2018)

관련 논문AI추천

A weaker notion of the finite factorization propertyHenry Jiang, Hwisoo Kim, Shihan Kanungo대한수학회 · 대한수학회논문집2024 · 제39권 제2호On the sets of lengths of Puiseux monoids generated by multiple geometric sequencesHarold Polo대한수학회 · 대한수학회논문집2020 · 제35권 제4호On the divisor-class group of monadic submonoids of rings of integer-valued polynomialsAndreas Reinhart대한수학회 · 대한수학회논문집2017 · 제32권 제2호Factorizations into irreducible numerical semigroupsPedro A. Garcıa-Sanchez대한수학회 · 대한수학회논문집2025 · 제40권 제3호The class group of D*/U for D an integral domain and U a group of units of D장규환강원경기수학회 · 한국수학논문집2009 · 제17권 제2호On the internal sum of Puiseux monoidsBryan Li, Jonathan Du, Shaohuan Zhang강원경기수학회 · 한국수학논문집2024 · 제32권 제4호

이 간행물의 다른 논문 (13건)

On a class of quasilinear elliptic equation with indefinite weights on graphsGuoqing Zhang, Shoudong Manpp. 857-867Homogeneity and symmetry on almost Kenmotsu $3$-manifoldsYaning Wangpp. 917-934Limit theorems for Hawkes processes with uniform immigrants설영수pp. 935-946Shifted tableau switchings and shifted Littlewood-Richardson coefficients최승일, 오영탁, 남선영pp. 947-984Diophantine inequality with four squares and one $k$th power of primesLi Zhupp. 985-1000
원문 보기