Chain conditions in special pullbacks

Jung Wook Lim; Dong Yeol Oh

doi:10.1016/j.crma.2012.07.003

Algebra

Chain conditions in special pullbacks

Presented by: Jean-Pierre Demailly

Jung Wook Lim ¹; Dong Yeol Oh ²

¹ Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea
² Division of Liberal Arts, Hanbat National University, Daejeon 305-719, Republic of Korea

Comptes Rendus. Mathématique, Volume 350 (2012) no. 13-14, pp. 655-659.

Abstract
Résumé

Let $D \subseteq E$ denote an extension of commutative rings with identity, I be a nonzero proper ideal of D, Γ mean a nonzero torsion-free additive grading monoid with $Γ \cap - Γ = {0}$ and $Γ^{⁎} = Γ ∖ {0}$ . Let $E [Γ]$ be the semigroup ring of Γ over E, $D + E [Γ^{⁎}] = {f \in E [Γ] | f (0) \in D}$ and $D + I [Γ^{⁎}] = {f \in D [Γ] |$ the coefficients of nonconstant terms of f belong to I}. In this paper, we give some conditions for the rings (resp., domains) $D + E [Γ^{⁎}]$ and $D + I [Γ^{⁎}]$ to be Noetherian (resp., to satisfy the ascending chain condition on principal ideals).

Soient $D \subseteq E$ une extension dʼanneaux commutatifs unitaires, I un idéal non nul et propre de D et Γ un monoïde commutatif simplifiable sans torsion non trivial tel que $Γ \cap - Γ = {0}$ et $Γ^{⁎} = Γ ∖ {0}$ . Soient $E [Γ]$ lʼanneau semi-groupe de Γ sur E, $D + E [Γ^{⁎}] = {f \in E [Γ] | f (0) \in D}$ et $D + I [Γ^{⁎}] = {f \in D [Γ] |$ les coefficients des termes non-constants de f appartiennent à I}. Dans cet article, nous donnons certaines conditions pour que les anneaux (resp., domaines) $D + E [Γ^{⁎}]$ et $D + I [Γ^{⁎}]$ soient Noethériens (resp., satisfassent la condition de chaîne ascendante sur les idéaux principaux).

Received: 2012-05-20
Accepted: 2012-07-05
Published online: 2012-07-01

DOI: 10.1016/j.crma.2012.07.003

Author's affiliations:

Jung Wook Lim ¹; Dong Yeol Oh ²

¹ Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea
² Division of Liberal Arts, Hanbat National University, Daejeon 305-719, Republic of Korea

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     author = {Jung Wook Lim and Dong Yeol Oh},
     title = {Chain conditions in special pullbacks},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {655--659},
     publisher = {Elsevier},
     volume = {350},
     number = {13-14},
     year = {2012},
     doi = {10.1016/j.crma.2012.07.003},
     language = {en},
}

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Jung Wook Lim; Dong Yeol Oh. Chain conditions in special pullbacks. Comptes Rendus. Mathématique, Volume 350 (2012) no. 13-14, pp. 655-659. doi : 10.1016/j.crma.2012.07.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.07.003/

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