Rocky Mountain Journal of Mathematics

The ascending chain condition on principal ideals in composite generalized power series rings

Jung Wook Lim and Dong Yeol Oh

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Abstract

Let $D \subseteq E$ be an extension of commutative rings with identity, $I$ a nonzero proper ideal of $D$, $(\Gamma , \leq )$ a strictly totally ordered monoid such that $0 \leq \alpha $ for all $\alpha \in \Gamma $, and $\Gamma ^*=\Gamma \setminus \{0\}$. Let $D+[\![E^{\Gamma ^*, \leq }]\!]=\{f \in [\![E^{\Gamma , \leq }]\!] \mid f(0) \in D\}$ and $D+[\![I^{\Gamma ^*, \leq }]\!] =\{f \in [\![D^{\Gamma , \leq }]\!] \mid f(\alpha ) \in I$ for all $\alpha \in \Gamma ^*\}$. In this paper, we give some conditions for the rings $D+[\![E^{\Gamma ^*, \leq }]\!]$ and $D+[\![I^{\Gamma ^*, \leq }]\!]$ to satisfy the ascending chain condition on principal ideals.

Article information

Source
Rocky Mountain J. Math., Volume 49, Number 4 (2019), 1223-1236.

Dates
First available in Project Euclid: 29 August 2019

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1567044037

Digital Object Identifier
doi:10.1216/RMJ-2019-49-4-1223

Mathematical Reviews number (MathSciNet)
MR3998919

Zentralblatt MATH identifier
07104715

Subjects
Primary: 13A02: Graded rings [See also 16W50] 13A15: Ideals; multiplicative ideal theory 13E99: None of the above, but in this section 13G05: Integral domains

Keywords
Generalized power series rings ring extensions ascending chain condition on principal ideals

Citation

Lim, Jung Wook; Oh, Dong Yeol. The ascending chain condition on principal ideals in composite generalized power series rings. Rocky Mountain J. Math. 49 (2019), no. 4, 1223--1236. doi:10.1216/RMJ-2019-49-4-1223. https://projecteuclid.org/euclid.rmjm/1567044037


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