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2018 The local $S$-class group of an integral domain
Ahmed Hamed
Rocky Mountain J. Math. 48(5): 1585-1605 (2018). DOI: 10.1216/RMJ-2018-48-5-1585

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]])?$

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Ahmed Hamed. "The local $S$-class group of an integral domain." Rocky Mountain J. Math. 48 (5) 1585 - 1605, 2018. https://doi.org/10.1216/RMJ-2018-48-5-1585

Information

Published: 2018
First available in Project Euclid: 19 October 2018

zbMATH: 06958793
MathSciNet: MR3866560
Digital Object Identifier: 10.1216/RMJ-2018-48-5-1585

Subjects:
Primary: 13A15, 13C20, 13F05, 13F25

Rights: Copyright © 2018 Rocky Mountain Mathematics Consortium

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Vol.48 • No. 5 • 2018
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