## Rocky Mountain Journal of Mathematics

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

#### 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

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

#### 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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