Every abelian group is the class group of a ring of Krull type
1
조회
0
다운로드
1
인용
0
참고문헌
키워드
초록
Let $Cl(A)$ denote the class group of an arbitrary integral domain $A$ introduced by Bouvier in 1982. Then $Cl(A)$ is the ideal class (resp., divisor class) group of $A$ if $A$ is a Dedekind or a Pr\"ufer (resp., Krull) domain. Let $G$ be an abelian group. In this paper, we show that there is a ring of Krull type $D$ such that $Cl(D) = G$ but $D$ is not a Krull domain. We then use this ring to construct a Pr\"ufer ring of Krull type $E$ such that $Cl(E) = G$ but $E$ is not a Dedekind domain. This is a generalization of Claborn's result that every abelian group is the ideal class group of a Dedekind domain.
Abstract
Let $Cl(A)$ denote the class group of an arbitrary integral domain $A$ introduced by Bouvier in 1982. Then $Cl(A)$ is the ideal class (resp., divisor class) group of $A$ if $A$ is a Dedekind or a Pr\"ufer (resp., Krull) domain. Let $G$ be an abelian group. In this paper, we show that there is a ring of Krull type $D$ such that $Cl(D) = G$ but $D$ is not a Krull domain. We then use this ring to construct a Pr\"ufer ring of Krull type $E$ such that $Cl(E) = G$ but $E$ is not a Dedekind domain. This is a generalization of Claborn's result that every abelian group is the ideal class group of a Dedekind domain.
참고문헌 (27)
- D.D. Anderson, "Splitting the t-class group", Journal of Pure and Applied Algebra, 74(1), pp.17~37 (1991), Elsevier BV[DOI]
- D.D. Anderson, "Integral domains of finite t -character", Journal of Algebra, 396, pp.169~183 (2013), Elsevier BV[DOI]
- David Anderson, "Some remarks on Prüfer ⋆-multiplication domains and class groups", Journal of Algebra, 319(1), pp.272~295 (2008), Elsevier BV[DOI]
- D.D. Anderson, "t-linked extensions, the t-class group, and Nagata's theorem", Journal of Pure and Applied Algebra, 86(2), pp.109~124 (1993), Elsevier BV[DOI]
- David F. Anderson, "The class group of D + M", Journal of Pure and Applied Algebra, 52(3), pp.199~212 (1988), Elsevier BV[DOI]
- Jimmy T. Arnold, "On the Contents of Polynomials", Proceedings of the American Mathematical Society, 24(3), pp.556~ (1970), JSTOR[DOI]
- A. Bouvier, "Le groupe des classes d’un anneau integre", 107 eme Congres des Societes Savantes, Brest fasc. IV, pp.85~92 (1982)
- A. Bouvier, "On some class groups of an integral domain", Bull. Soc. Math. Gr`ece (N.S.), 29, pp.45~59 (1988)
- Gyu Whan Chang, "Integral Domains in which Every Nonzero t -Locally Principal Ideal is t -Invertible", Communications in Algebra, 41(10), pp.3805~3819 (2013), Informa UK Limited[DOI]
- L. Claborn, "Every abelian group is a class group", Pacific J. Math., 18, pp.219~222 (1966)
- "Journal of Algebra", 1978, 423~439, pp.Elsevier BV, 10.1016[DOI]
- Edward D. Davis, "Overrings of Commutative Rings. II. Integrally Closed Overrings", Transactions of the American Mathematical Society, 110(2), pp.196~ (1964), JSTOR[DOI]
- David E. Dobbs, "t-Linked overrings and prüfer v-multiplication domains", Communications in Algebra, 17(11), pp.2835~2852 (1989), Informa UK Limited[DOI]
- S. El Baghdadi, "On the class group of a graded domain", J. Pure Appl. Algebra, 171(2-3), pp.171~184 (2002)
- M. Fontana, "UMT-domains and domains with Pr¨ufer integral closure, Comm", Algebra, 26(4), pp.1017~1039 (1998)[DOI]
- R. M. Fossum, "The Divisor Class Group of a Krull Domain" (1973), Springer-Verlag
- R. Gilmer, "Multiplicative Ideal Theory" (1972), Marcel Dekker, Inc
- Malcolm Griffin, "Some Results on v-Multiplication Rings", Canadian Journal of Mathematics, 19, pp.710~722 (1967), Canadian Mathematical Society[DOI]
- M. Griffin, "Rings of Krull type", Journal für die reine und angewandte Mathematik (Crelles Journal), 1968(229), pp.1~27 (1968), Walter de Gruyter GmbH[DOI]
- John R. Hedstorm, "Some remarks on star-operations", Journal of Pure and Applied Algebra, 18(1), pp.37~44 (1980), Elsevier BV[DOI]
- William Heinzer, "Integral domains in which each non‐zero ideal is divisorial", Mathematika, 15(2), pp.164~170 (1968), Wiley[DOI]
- E. Houston, "Integral domains in which each t-ideal is divisorial", Michigan Math. J, 35(2), pp.291~300 (1988)[DOI]
- Evan Houston, "On t-invertibility II", Communications in Algebra, 17(8), pp.1955~1969 (1989), Informa UK Limited[DOI]
- Nv, "Journal of Algebra", 1989, 151~170, pp.Elsevier BV, 10.1016[DOI]
- B.G Kang, "On the converse of a well-known fact about Krull domains", Journal of Algebra, 124(2), pp.284~299 (1989), Elsevier BV[DOI]
- S. Malik, "Algebra and its applications", pp.133~146 (1981), Dekker
- Joe L. Mott, "On Prufer v-multiplication domains", Manuscripta Mathematica, 35(1-2), pp.1~26 (1981), Springer Science and Business Media LLC[DOI]