Composite Hurwitz rings satisfying the ascending chain condition on principal ideals

Jung Wook Lim; Dong Yeol Oh

doi:10.5666/KMJ.2016.56.4.1115

Composite Hurwitz rings satisfying the ascending chain condition on principal ideals

Jung Wook Lim, Dong Yeol Oh

Chosun University

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

8 Scopus citations

Abstract

Let D ⊆ E be an extension of integral domains with characteristic zero, I be a nonzero proper ideal of D and let H(D,E) and H(D,I) (resp., h(D,E) and h(D,I)) be composite Hurwitz series rings (resp., composite Hurwitz polynomial rings). In this paper, we show that H(D,E) satisfies the ascending chain condition on principal ideals if and only n if h(D,E) satisfies the ascending chain condition on principal ideals, if and only if ∩ _n≥1 a₁⋯a_nE = (0) for each infinite sequence (a_n)_n≥1 consisting of nonzero nonunits of D. We also prove that H(D,I) satisfies the ascending chain condition on principal ideals if and only if h(D,I) satisfies the ascending chain condition on principal ideals, if and only if D satisfies the ascending chain condition on principal ideals.

Original language	English
Pages (from-to)	1115-1123
Number of pages	9
Journal	Kyungpook Mathematical Journal
Volume	56
Issue number	4
DOIs	https://doi.org/10.5666/KMJ.2016.56.4.1115
State	Published - 2016

Keywords

Ascending chain condition on principal ideals
Composite Hurwitz polynomial ring
Composite Hurwitz series ring

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title = "Composite Hurwitz rings satisfying the ascending chain condition on principal ideals",

abstract = "Let D ⊆ E be an extension of integral domains with characteristic zero, I be a nonzero proper ideal of D and let H(D,E) and H(D,I) (resp., h(D,E) and h(D,I)) be composite Hurwitz series rings (resp., composite Hurwitz polynomial rings). In this paper, we show that H(D,E) satisfies the ascending chain condition on principal ideals if and only n if h(D,E) satisfies the ascending chain condition on principal ideals, if and only if ∩ n≥1 a1⋯anE = (0) for each infinite sequence (an)n≥1 consisting of nonzero nonunits of D. We also prove that H(D,I) satisfies the ascending chain condition on principal ideals if and only if h(D,I) satisfies the ascending chain condition on principal ideals, if and only if D satisfies the ascending chain condition on principal ideals.",

keywords = "Ascending chain condition on principal ideals, Composite Hurwitz polynomial ring, Composite Hurwitz series ring",

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

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T1 - Composite Hurwitz rings satisfying the ascending chain condition on principal ideals

AU - Lim, Jung Wook

AU - Oh, Dong Yeol

PY - 2016

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N2 - Let D ⊆ E be an extension of integral domains with characteristic zero, I be a nonzero proper ideal of D and let H(D,E) and H(D,I) (resp., h(D,E) and h(D,I)) be composite Hurwitz series rings (resp., composite Hurwitz polynomial rings). In this paper, we show that H(D,E) satisfies the ascending chain condition on principal ideals if and only n if h(D,E) satisfies the ascending chain condition on principal ideals, if and only if ∩ n≥1 a1⋯anE = (0) for each infinite sequence (an)n≥1 consisting of nonzero nonunits of D. We also prove that H(D,I) satisfies the ascending chain condition on principal ideals if and only if h(D,I) satisfies the ascending chain condition on principal ideals, if and only if D satisfies the ascending chain condition on principal ideals.

AB - Let D ⊆ E be an extension of integral domains with characteristic zero, I be a nonzero proper ideal of D and let H(D,E) and H(D,I) (resp., h(D,E) and h(D,I)) be composite Hurwitz series rings (resp., composite Hurwitz polynomial rings). In this paper, we show that H(D,E) satisfies the ascending chain condition on principal ideals if and only n if h(D,E) satisfies the ascending chain condition on principal ideals, if and only if ∩ n≥1 a1⋯anE = (0) for each infinite sequence (an)n≥1 consisting of nonzero nonunits of D. We also prove that H(D,I) satisfies the ascending chain condition on principal ideals if and only if h(D,I) satisfies the ascending chain condition on principal ideals, if and only if D satisfies the ascending chain condition on principal ideals.

KW - Ascending chain condition on principal ideals

KW - Composite Hurwitz polynomial ring

KW - Composite Hurwitz series ring

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JO - Kyungpook Mathematical Journal

JF - Kyungpook Mathematical Journal

IS - 4

ER -

Composite Hurwitz rings satisfying the ascending chain condition on principal ideals

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Keywords

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