When is $ A+XB[[X]]$ Noetherian?

Sana Hizem; Ali Benhissi

doi:10.1016/j.crma.2004.11.017

Algebra

When is

A + X B [[X]]

Noetherian?

Presented by: Jean-Pierre Demailly

Sana Hizem ¹; Ali Benhissi ¹

¹ Department of Mathematics, Faculty of Sciences, 5000 Monastir, Tunisia

Comptes Rendus. Mathématique, Volume 340 (2005) no. 1, pp. 5-7.

Abstract
Résumé

Let $A \subseteq B$ be an extension of commutative rings with identity, X an analytic indeterminate over B, and $R : = A + X B [[X]]$ , the subring of the formal power series ring $B [[X]]$ , consisting of the series with constant terms in A. In this Note we study when the ring R is Noetherian. We prove that R is Noetherian if and only if A is Noetherian and B is a finitely generated A-module.

Soient $A \subseteq B$ une extension d'anneaux commutatifs unitaires, X une indeterminée sur B, et $R : = A + X B [[X]]$ , le sous-anneau de l'anneau des séries formelles $B [[X]]$ , formé par les séries dont le terme constant est dans A. Nous donnons une condition nécessaire et suffisante pour que l'anneau R soit noethérien. Nous démontrons que R est noethérien si et seulement si A est noethérien et B est un A module de type fini.

Received: 2004-05-04
Accepted: 2004-11-05
Published online: 2004-12-22

DOI: 10.1016/j.crma.2004.11.017

Author's affiliations:

Sana Hizem ¹; Ali Benhissi ¹

¹ Department of Mathematics, Faculty of Sciences, 5000 Monastir, Tunisia

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     title = {When is $ A+XB[[X]]$ {Noetherian?}},
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Sana Hizem; Ali Benhissi. When is $ A+XB[[X]]$ Noetherian?. Comptes Rendus. Mathématique, Volume 340 (2005) no. 1, pp. 5-7. doi : 10.1016/j.crma.2004.11.017. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.11.017/

References
Cited by

[1] D.F. Anderson; D.N. El Abidine The $A + X B [X]$ and $A + X B [[X]]$ constructions from GCD-domains, J. Pure Appl. Algebra, Volume 159 (2001), pp. 15-24

[2] D.E. Dobbs; M. Khalis On the prime spectrum, Krull dimension and catenarity of integral domains of the form $A + X B [[X]]$ , J. Pure Appl. Algebra, Volume 159 (2001), pp. 57-73

[3] T. Dumitrescu; S.O.I. Al-Salihi; N. Radu; T. Shah Some factorization properties of composite domains $A + X B [X]$ and $A + X B [[X]]$ , Commun. Algebra, Volume 28 (2000) no. 3, pp. 1125-1139

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