On the cardinality of stable star operations of finite type on an integral domain

Gyu Whan Chang

doi:10.1016/j.crma.2012.05.015

Algebra

On the cardinality of stable star operations of finite type on an integral domain

Presented by: the Editorial Board

Gyu Whan Chang ¹

¹ Department of Mathematics, University of Incheon, Incheon 406-772, Republic of Korea

Comptes Rendus. Mathématique, Volume 350 (2012) no. 11-12, pp. 557-560.

Abstract
Résumé

Let D be an integral domain and ${SF}_{s} (D)$ be the set of stable star operations of finite type on D. In this note, we show that if Ω is the set of nonzero prime ideals P of D with $P^{t} = D$ , then $| Ω | + 1 ⩽ | {SF}_{s} (D) | ⩽ 2^{| Ω |}$ . We also show that if $| Ω | < \infty$ , then $| {SF}_{s} (D) | = | Ω | + 1$ if and only if Ω is linearly ordered under inclusion; and $| {SF}_{s} (D) | = 2^{| Ω |}$ if and only if each pair of elements in Ω are incomparable.

Soit D un anneau intègre et ${SF}_{s} (D)$ lʼensemble des opérations étoile, stables, de type fini sur D. Nous montrons dans cette note que, si Ω désigne lʼensemble des idéaux premiers non nuls P de D tels que $P^{t} = D$ , alors $| Ω | + 1 ⩽ | {SF}_{s} (D) | ⩽ 2^{| Ω |}$ . Nous montrons également que, si $| Ω | < \infty$ , alors $| {SF}_{s} (D) | = | Ω | + 1$ si et seulement si Ω est totalement ordonné par lʼinclusion et $| {SF}_{s} (D) | = 2^{| Ω |}$ si et seulement si les éléments de Ω sont deux à deux incomparables.

Received: 2012-03-09
Accepted: 2012-05-29
Published online: 2012-06-01

DOI: 10.1016/j.crma.2012.05.015

Author's affiliations:

Gyu Whan Chang ¹

¹ Department of Mathematics, University of Incheon, Incheon 406-772, Republic of Korea

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     title = {On the cardinality of stable star operations of finite type on an integral domain},
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Gyu Whan Chang. On the cardinality of stable star operations of finite type on an integral domain. Comptes Rendus. Mathématique, Volume 350 (2012) no. 11-12, pp. 557-560. doi : 10.1016/j.crma.2012.05.015. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.05.015/

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