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

Jung Wook Lim; Dong Yeol Oh

doi:10.1216/RMJ-2019-49-4-1223

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

Jung Wook Lim
, Dong Yeol Oh

Chosun University

Research output: Contribution to journal › Article › peer-review

Abstract

Let D ⊆ E be an extension of commutative rings with identity, I a nonzero proper ideal of D, (γ;≤) a strictly totally ordered monoid such that 0 ≤αfor all α ∈ γ, and γ∗ = γ\ {0}. Let D+[Eγ∗, ≤] = {f 2 [Eγ;≤] f(0) ∈ D} and D+[Iγ∗, ≤] = {f ∈ [Dγ;≤] f(α) ∈ I for all α ∈ γ}. In this paper, we give some conditions for the rings D+[Eγ∗, ≤] and D +[Iγ∗, ≤] to satisfy the ascending chain condition on principal ideals.

Original language	English
Pages (from-to)	1223-1236
Number of pages	14
Journal	Rocky Mountain Journal of Mathematics
Volume	49
Issue number	4
DOIs	https://doi.org/10.1216/RMJ-2019-49-4-1223
State	Published - 2019

Keywords

Ascending chain condition on principal ideals
Generalized power series rings
Ring extensions

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10.1216/RMJ-2019-49-4-1223

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title = "The ascending chain condition on principal ideals in composite generalized power series rings",

abstract = "Let D ⊆ E be an extension of commutative rings with identity, I a nonzero proper ideal of D, (γ;≤) a strictly totally ordered monoid such that 0 ≤αfor all α ∈ γ, and γ∗ = γ\textbackslash{} \{0\}. Let D+[Eγ∗, ≤] = \{f 2 [Eγ;≤] f(0) ∈ D\} and D+[Iγ∗, ≤] = \{f ∈ [Dγ;≤] f(α) ∈ I for all α ∈ γ\}. In this paper, we give some conditions for the rings D+[Eγ∗, ≤] and D +[Iγ∗, ≤] to satisfy the ascending chain condition on principal ideals.",

keywords = "Ascending chain condition on principal ideals, Generalized power series rings, Ring extensions",

author = "Lim, \{Jung Wook\} and Oh, \{Dong Yeol\}",

note = "Publisher Copyright: {\textcopyright} 2019 Rocky Mountain Mathematics Consortium.",

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doi = "10.1216/RMJ-2019-49-4-1223",

language = "English",

volume = "49",

pages = "1223--1236",

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T1 - The ascending chain condition on principal ideals in composite generalized power series rings

AU - Lim, Jung Wook

AU - Oh, Dong Yeol

PY - 2019

Y1 - 2019

N2 - Let D ⊆ E be an extension of commutative rings with identity, I a nonzero proper ideal of D, (γ;≤) a strictly totally ordered monoid such that 0 ≤αfor all α ∈ γ, and γ∗ = γ\ {0}. Let D+[Eγ∗, ≤] = {f 2 [Eγ;≤] f(0) ∈ D} and D+[Iγ∗, ≤] = {f ∈ [Dγ;≤] f(α) ∈ I for all α ∈ γ}. In this paper, we give some conditions for the rings D+[Eγ∗, ≤] and D +[Iγ∗, ≤] to satisfy the ascending chain condition on principal ideals.

AB - Let D ⊆ E be an extension of commutative rings with identity, I a nonzero proper ideal of D, (γ;≤) a strictly totally ordered monoid such that 0 ≤αfor all α ∈ γ, and γ∗ = γ\ {0}. Let D+[Eγ∗, ≤] = {f 2 [Eγ;≤] f(0) ∈ D} and D+[Iγ∗, ≤] = {f ∈ [Dγ;≤] f(α) ∈ I for all α ∈ γ}. In this paper, we give some conditions for the rings D+[Eγ∗, ≤] and D +[Iγ∗, ≤] to satisfy the ascending chain condition on principal ideals.

KW - Ascending chain condition on principal ideals

KW - Generalized power series rings

KW - Ring extensions

UR - https://www.scopus.com/pages/publications/85072667010

U2 - 10.1216/RMJ-2019-49-4-1223

DO - 10.1216/RMJ-2019-49-4-1223

M3 - Article

AN - SCOPUS:85072667010

SN - 0035-7596

VL - 49

SP - 1223

EP - 1236

JO - Rocky Mountain Journal of Mathematics

JF - Rocky Mountain Journal of Mathematics

IS - 4

ER -

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

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