Rocky Mountain Journal of Mathematics

The local $S$-class group of an integral domain

Ahmed Hamed

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Abstract

In this paper, we define the local $S$-class group of an integral domain $D$. A nonzero fractional ideal $I$ of $D$ is said to be $S$-invertible if there exist an $s\in S$ and a fractional ideal $J$ of $D$ such that $sD \subseteq I, J \subseteq D$. The local $S$-class group of $D$, denoted G$ (D)$, is the group of fractional $t$-invertible $t$-ideals of $D$ under $t$-multiplication modulo its subgroup of $S$-invertible $t$-invertible $t$-ideals of $D$. We study the case {G }$(D)=0$, and we generalize some known results developed for the classic contexts of Krull and P$\upsilon $MD domains. Moreover, we investigate the case of isomorphism {G }$(D) \simeq$ {G }$(D[[X]])$. In particular, we give with an additional condition an answer to the question of Bouvier, that is, when is G$ (D)$ isomorphic to G$ (D[[X]])?$

Article information

Source
Rocky Mountain J. Math., Volume 48, Number 5 (2018), 1585-1605.

Dates
First available in Project Euclid: 19 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1539936037

Digital Object Identifier
doi:10.1216/RMJ-2018-48-5-1585

Mathematical Reviews number (MathSciNet)
MR3866560

Zentralblatt MATH identifier
06958793

Subjects
Primary: 13A15: Ideals; multiplicative ideal theory 13C20: Class groups [See also 11R29] 13F05: Dedekind, Prüfer, Krull and Mori rings and their generalizations 13F25: Formal power series rings [See also 13J05]

Keywords
Local class group formal power series $S$-invertible ideal

Citation

Hamed, Ahmed. The local $S$-class group of an integral domain. Rocky Mountain J. Math. 48 (2018), no. 5, 1585--1605. doi:10.1216/RMJ-2018-48-5-1585. https://projecteuclid.org/euclid.rmjm/1539936037


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References

  • D.D. Anderson, $\pi$-domains, divisorial ideals and overrings, Glasgow Math. J. 19 (1978), 199–203.
  • D.D. Anderson and D.F. Anderson, Generalized GCD-domains, Comm. Math. Univ. 28 (1979), 215–221.
  • ––––, Some remarks on star operations and the class group, J. Pure Appl. Alg. 51 (1988), 27–33.
  • D.D. Anderson and T. Dumitrescu, $S$-Noetherian rings, Comm. Algebra 30 (2002), 4407–4416.
  • D.D. Anderson, E.G. Houston and M. Zafrullah, $t$-linked extensions, the $t$-class group, and Nagat's theorem, J. Pure Appl. Algebra 86 (1993), 109–129.
  • D.F. Anderson, S.E. Baghdadi and S.E. Kabbaj, On the class group of $A+XB[X]$ domains, Lect. Notes Pure Appl. Math. 205 (1999), 73–85.
  • A. Bouvier, Le groupe des classes d'un anneau intègre, Cong. Soc. Savantes 107 (1982), 85–92.
  • ––––, The local class group of a Krull domain, Canadian Math. Bull. 26 (1983), 13–19.
  • A. Bouvier and M. Zafrullah, On some class groups of an integral domain, Bull. Greek Math. Soc. 29 (1988), 45–59.
  • S. Gabelli, On divisorial ideal in polynomal rings over Mori domains, Comm. Algebra 15 (1987), 2349–2370.
  • R. Gilmer, Multiplicative ideal theory, Marcel Dekker, New York, 1972.
  • A. Hamed and S. Hizem, On the class group and $S$-class group of formal power series rings, J. Pure Appl. Algebra 221 (2017), 2869–2879.
  • E. Houston and M. Zafrullah, Integral domains in which each $t$-ideal is divisorial, Michigan Math. J. 35 (1988), 291–300.
  • B.G. Kang, Prüfer $v$-multiplication domains and the ring $R[X]_{N_v}$, J. Algebra 123 (1989), 151–170.
  • M. Zafrullah, On generalized Dedekind domains, Mathematika 33 (1986), 285–295.
  • M. Zafrullah, On a property of pre-Schreier domains, Comm. Algebra 15 (1987), 1895–1920.