## Rocky Mountain Journal of Mathematics

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

Ahmed Hamed

#### 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

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

#### 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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