Rocky Mountain Journal of Mathematics

Weakly factorial property of a generalized Rees ring $D[X,d/X]$

Gyu Whan Chang

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Abstract

Let $D$ be an integral domain, $X$ an indeterminate over $D$, $d \in D$, and $R = D[X, {d}/{X}]$ a subring of $D[X,{1}/{X}]$. In this paper, we show that $R$ is a weakly factorial domain if and only if $D$ is a weakly factorial GCD-domain and $d=0$, $d$ is a unit of $D$ or $d$ is a prime element of $D$. We also show that, if $D$ is a weakly factorial GCD-domain, $p$ is a prime element of $D$, and $n \geq 2$ is an integer, then $D[X, {p^n}/{X}]$ is an almost weakly factorial domain with $Cl(D[X, {p^n}/{X}]) = \mathbb {Z}_n$.

Article information

Source
Rocky Mountain J. Math., Volume 48, Number 7 (2018), 2175-2185.

Dates
First available in Project Euclid: 14 December 2018

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1544756806

Digital Object Identifier
doi:10.1216/RMJ-2018-48-7-2175

Mathematical Reviews number (MathSciNet)
MR3892129

Zentralblatt MATH identifier
06999259

Subjects
Primary: 13A15: Ideals; multiplicative ideal theory 13F15: Rings defined by factorization properties (e.g., atomic, factorial, half- factorial) [See also 13A05, 14M05]

Keywords
Weakly factorial domain GCD-domain generalized Rees ring $D[X d/X]$

Citation

Chang, Gyu Whan. Weakly factorial property of a generalized Rees ring $D[X,d/X]$. Rocky Mountain J. Math. 48 (2018), no. 7, 2175--2185. doi:10.1216/RMJ-2018-48-7-2175. https://projecteuclid.org/euclid.rmjm/1544756806


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