## Rocky Mountain Journal of Mathematics

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

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

#### 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