Hostname: page-component-7dd5485656-glrdx Total loading time: 0 Render date: 2025-10-29T09:03:10.487Z Has data issue: false hasContentIssue false
  • English
  • Français

Revisiting G-Dedekind domains

Part of: General commutative ring theory Arithmetic rings and other special rings Integral domains

Published online by Cambridge University Press:  20 December 2021

M. Zafrullah Open the ORCID record for M. Zafrullah [Opens in a new window]
M. Zafrullah*
Affiliation:
Department of Mathematics, Idaho State University, Pocatello, ID 83209, USA
*
e-mail: mzafrullah@usa.net
Get access Rights & Permissions [Opens in a new window]

Abstract

Let R be an integral domain with $qf(R)=K$, and let $F(R)$ be the set of nonzero fractional ideals of R. Call R a dually compact domain (DCD) if, for each $I\in F(R)$, the ideal $I_{v}=(I^{-1})^{-1}$ is a finite intersection of principal fractional ideals. We characterize DCDs and show that the class of DCDs properly contains various classes of integral domains, such as Noetherian, Mori, and Krull domains. In addition, we show that a Schreier DCD is a greatest common divisor (GCD) (Greatest Common Divisor) domain with the property that, for each $A\in F(R)$, the ideal $A_{v}$ is principal. We show that a domain R is G-Dedekind (i.e., has the property that $A_{v}$ is invertible for each $A\in F(R)$) if and only if R is a DCD satisfying the property $\ast :$ For all pairs of subsets $\{a_{1},\ldots ,a_{m}\},\{b_{1},\ldots ,b_{n}\}\subseteq K\backslash \{0\}, (\cap _{i=1}^{m}(a_{i})(\cap _{j=1}^{n}(b_{j}))=\cap _{i,j=1}^{m,n}(a_{i}b_{j})$. We discuss what the appropriate names for G-Dedekind domains and related notions should be. We also make some observations about how the DCDs behave under localizations and polynomial ring extensions.

Keywords

G-DedekindPseudo-Dedekindinvertible*-invertibleNoetherianMori

MSC classification

Primary: 13A15: Ideals; multiplicative ideal theory
Secondary: 13A05: Divisibility; factorizations 13F05: Dedekind, Prüfer, Krull and Mori rings and their generalizations 13G05: Integral domains

Information

Type
Article
Information
Canadian Mathematical Bulletin , Volume 66 , Issue 1 , March 2023 , pp. 38 - 52
Check for updates
Copyright
© Canadian Mathematical Society, 2021

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

Dedicated to fairness

References

Akalan, E., On generalized Dedekind prime rings . J. Algebra 320(2008), 2907–2916.CrossRefGoogle Scholar
Anderson, D. D., On the ideal equation $I\left(B\cap C\right)= IB\cap IC$ . Canad. Math. Bull. 26(1983), 331–332.CrossRefGoogle Scholar
Anderson, D. D. and Anderson, D. F., Generalized GCD domains . Comment. Math. Univ. St. Pauli. 28(1979), 215–221.Google Scholar
Anderson, D. D., Anderson, D. F., Fontana, M., and Zafrullah, M., On $\textit{v}$ -domains and star operations . Comm. Algebra. 37(2009), 3018–3043.CrossRefGoogle Scholar
Anderson, D. D. and Cook, S., Two star operations and their induced lattices . Comm. Algebra. 28(2000), no. 5, 2461–2475.CrossRefGoogle Scholar
Anderson, D. D. and Kang, B. G., Pseudo-Dedekind domains and divisorial ideals in R[X]T . J. Algebra 122(1989), 323–336.CrossRefGoogle Scholar
Anderson, D. D., Kwak, D. J., and Zafrullah, M., Agreeable domains . Comm. Algebra. 23(2010), no. 13, 4861–4883.CrossRefGoogle Scholar
Anderson, D. F. and Dobbs, D., On the product of ideals . Canad. Math. Bull. 26(1983), 106–116.CrossRefGoogle Scholar
Cohn, P. M., Bézout rings and their subrings . Proc. Camb. Philos. Soc. 64(1968), 251–264.CrossRefGoogle Scholar
Elliott, J., Rings, modules and closure operations, Springer Monographs in Mathematics, Part of the Springer Monographs in Mathematics book series (SMM). Springer, Cham, 2019.CrossRefGoogle Scholar
Fontana, M. and Gabelli, S., On the class group and the local class group of a pullback . J. Algebra 181(1996), 803–835.CrossRefGoogle Scholar
Fontana, M., Jara, P., and Santos, E., Prüfer $\star$ -multiplication domains and semistar operations . J. Algebra Appl. 2 (2003) 21–50.CrossRefGoogle Scholar
Fossum, R., The divisor class group of a Krull domain, Ergebnisse der Mathematik und ihrer grenzgebiete B, 74, Springer, Berlin–Heidelberg–New York, 1973.CrossRefGoogle Scholar
Gilmer, R. W., Multiplicative ideal theory, Marcel Dekker, New York, 1972.Google Scholar
Halter-Koch, F., Ideal systems: an introduction to multiplicative ideal theory, Marcel Dekker, New York, 1998.Google Scholar
Houston, E., On divisorial prime ideals in Prüfer v-multiplication domains . J. Pure Appl. Algebra 42(1986), 55–62.CrossRefGoogle Scholar
Houston, E., Malik, S., and Mott, J., Characterizations of $\star$ -multiplication domains , Canad. Math. Bull. 27(1984), no. 1, 48–52.CrossRefGoogle Scholar
Huckaba, J. and Papick, I., When the dual of an ideal is a ring . Manuscripta Math. 37(1982), 67–85.CrossRefGoogle Scholar
Jaffard, P., Les systemes d’ideaux, Dunod, Paris, 1960.Google Scholar
Popescu, N., On a class of Prüfer domains . Rev. Roumaine Math. Pure Appl. 29(1984), 777–786.Google Scholar
Querre, J., Ideaux divisoriels d’un anneau de polynomes . J. Algebra 64(1980), 270–284.CrossRefGoogle Scholar
Roitman, M., On polynomial extensions of Mori domains over countable fields . J. Pure Appl. Algebra 64(1990), 315–328.CrossRefGoogle Scholar
Wang, F. and McCasland, R., On w-modules over strong Mori domains . Comm. Algebra 25(1997), 1285–1306.Google Scholar
Zafrullah, M., Finite conductor domains . Manuscripta Math. 24(1978), 191–203.CrossRefGoogle Scholar
Zafrullah, M., On generalized Dedekind domains . Mathematika 33(1986), 285–295.CrossRefGoogle Scholar
Zafrullah, M., On a property of pre-Schreier domains . Comm. Alg. 15(1987), 1895–1920.CrossRefGoogle Scholar
Zafrullah, M., Ascending chain conditions and star operations . Comm. Algebra 17(1989), 1523–1533.CrossRefGoogle Scholar
Zafrullah, M., Chapter 20: putting t-invertibility to use. In: Glaz, S. and Chapman, S. (eds.), Non-Noetherian commutative ring theory, Mathematics and Its Applications, 520, Kluwer Academic, Dordrecht, Netherlands, 2000, pp. 429–457.CrossRefGoogle Scholar