Algebra
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.

Let D be an integral domain and SFs(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 Pt=D, then |Ω|+1|SFs(D)|2|Ω|. We also show that if |Ω|<, then |SFs(D)|=|Ω|+1 if and only if Ω is linearly ordered under inclusion; and |SFs(D)|=2|Ω| if and only if each pair of elements in Ω are incomparable.

Soit D un anneau intègre et SFs(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 Pt=D, alors |Ω|+1|SFs(D)|2|Ω|. Nous montrons également que, si |Ω|<, alors |SFs(D)|=|Ω|+1 si et seulement si Ω est totalement ordonné par lʼinclusion et |SFs(D)|=2|Ω| si et seulement si les éléments de Ω sont deux à deux incomparables.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2012.05.015
Chang, Gyu Whan 1

1 Department of Mathematics, University of Incheon, Incheon 406-772, Republic of Korea
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Chang, Gyu Whan. 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. http://www.numdam.org/articles/10.1016/j.crma.2012.05.015/

[1] Anderson, D.D. Star-operations induced by overrings, Comm. Algebra, Volume 16 (1988), pp. 2535-2553

[2] Anderson, D.D.; Cook, S.J. Two star-operations and their induced lattices, Comm. Algebra, Volume 28 (2000), pp. 2461-2475

[3] Dobbs, D.E.; Houston, E.G.; Lucas, T.G.; Roitman, M.; Zafrullah, M. On t-linked overrings, Comm. Algebra, Volume 20 (1992), pp. 1463-1488

[4] Dobbs, D.E.; Houston, E.G.; Lucas, T.G.; Zafrullah, M. t-linked overrings and Prüfer v-multiplication domains, Comm. Algebra, Volume 17 (1989), pp. 2835-2852

[5] Gilmer, R. Multiplicative Ideal Theory, Queenʼs Papers in Pure Appl. Math., vol. 90, Queenʼs University, Kingston, ON, Canada, 1992

[6] Houston, E.; Mimouni, A.; Park, M.H. Integral domains which admit at most two star operations, Comm. Algebra, Volume 39 (2011), pp. 1907-1921

[7] Houston, E.; Zafrullah, M. On t-invertibility II, Comm. Algebra, Volume 17 (1989), pp. 1955-1969

[8] Mimouni, A. Integral domains in which each ideal is a w-ideal, Comm. Algebra, Volume 33 (2005), pp. 1345-1355

[9] Picozza, G.; Tartarone, F. When the semistar operation is the identity, Comm. Algebra, Volume 36 (2008), pp. 1954-1975

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