Comptes Rendus
Algebra
Chain conditions in special pullbacks
Comptes Rendus. Mathématique, Volume 350 (2012) no. 13-14, pp. 655-659.

Let DE 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 ΓΓ={0} and Γ=Γ{0}. Let E[Γ] be the semigroup ring of Γ over E, D+E[Γ]={fE[Γ]|f(0)D} and D+I[Γ]={fD[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 DE 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 ΓΓ={0} et Γ=Γ{0}. Soient E[Γ] lʼanneau semi-groupe de Γ sur E, D+E[Γ]={fE[Γ]|f(0)D} et D+I[Γ]={fD[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).

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Published online:
DOI: 10.1016/j.crma.2012.07.003

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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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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