Rocky Mountain Journal of Mathematics

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

Ahmed Hamed

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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

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

First available in Project Euclid: 19 October 2018

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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]

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


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.

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