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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.
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참고문헌 (27)
참고문헌 신청
D.D. Anderson / 1991 / Splitting the t-class group / Journal of Pure and Applied Algebra 74(1):17~37
D.D. Anderson / 2013 / Integral domains of finite t -character / Journal of Algebra 396:169~183
David Anderson / 2008 / Some remarks on Prüfer ⋆-multiplication domains and class groups / Journal of Algebra 319(1):272~295
D.D. Anderson / 1993 / t-linked extensions, the t-class group, and Nagata's theorem / Journal of Pure and Applied Algebra 86(2):109~124
David F. Anderson / 1988 / The class group of D + M / Journal of Pure and Applied Algebra 52(3):199~212
D.D. Anderson / 2013 / Integral domains of finite t -character / Journal of Algebra 396:169~183
David Anderson / 2008 / Some remarks on Prüfer ⋆-multiplication domains and class groups / Journal of Algebra 319(1):272~295
D.D. Anderson / 1993 / t-linked extensions, the t-class group, and Nagata's theorem / Journal of Pure and Applied Algebra 86(2):109~124
David F. Anderson / 1988 / The class group of D + M / Journal of Pure and Applied Algebra 52(3):199~212
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