The $ D+E[{\Gamma }^{⁎}]$ construction from Prüfer domains and GCD-domains

Jung Wook Lim

doi:10.1016/j.crma.2011.10.023

Algebra

The

D + E [Γ^{⁎}]

construction from Prüfer domains and GCD-domains
[Construction

D + E [Γ^{⁎}]

dʼanneaux de Prüfer et à pgcd]

Présenté par : Jean-Pierre Demailly

Jung Wook Lim ¹

¹ Department of Mathematics, Pohang University of Science and Technology, Pohang 790-784, Republic of Korea

Comptes Rendus. Mathématique, Volume 349 (2011) no. 21-22, pp. 1135-1138.

Résumé
Abstract

Soit $D ⊊ E$ une extension dʼanneaux commutatifs intègres, Γ un monoïde commutatif simplifiable sans torsion non trivial tel que $Γ \cap - Γ = {0}$ . On note $Γ^{⁎} = Γ ∖ {0}$ et soit $D + E [Γ^{⁎}] = {f \in E [Γ] | f (0) \in D}$ . Dans cette note, on donne des conditions nécessaires et suffisantes pour que $D + E [Γ^{⁎}]$ soit un anneau de Prüfer ou un anneau à pgcd.

Let $D ⊊ E$ denote an extension of integral domains, Γ be a nonzero torsion-free grading monoid with $Γ \cap - Γ = {0}$ , $Γ^{⁎} = Γ ∖ {0}$ and $D + E [Γ^{⁎}] = {f \in E [Γ] | f (0) \in D}$ . In this paper, we give a necessary and sufficient criteria for $D + E [Γ^{⁎}]$ to be a Prüfer domain or a GCD-domain.

Reçu le : 2011-08-12
Accepté le : 2011-10-24
Publié le : 2011-11-01

DOI : 10.1016/j.crma.2011.10.023

Affiliations des auteurs :

Jung Wook Lim ¹

¹ Department of Mathematics, Pohang University of Science and Technology, Pohang 790-784, Republic of Korea

@article{CRMATH_2011__349_21-22_1135_0,
     author = {Jung Wook Lim},
     title = {The $ D+E[{\Gamma }^{{\textasteriskcentered}}]$ construction from {Pr\"ufer} domains and {GCD-domains}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1135--1138},
     publisher = {Elsevier},
     volume = {349},
     number = {21-22},
     year = {2011},
     doi = {10.1016/j.crma.2011.10.023},
     language = {en},
}

TY  - JOUR
AU  - Jung Wook Lim
TI  - The $ D+E[{\Gamma }^{⁎}]$ construction from Prüfer domains and GCD-domains
JO  - Comptes Rendus. Mathématique
PY  - 2011
SP  - 1135
EP  - 1138
VL  - 349
IS  - 21-22
PB  - Elsevier
DO  - 10.1016/j.crma.2011.10.023
LA  - en
ID  - CRMATH_2011__349_21-22_1135_0
ER  -

%0 Journal Article
%A Jung Wook Lim
%T The $ D+E[{\Gamma }^{⁎}]$ construction from Prüfer domains and GCD-domains
%J Comptes Rendus. Mathématique
%D 2011
%P 1135-1138
%V 349
%N 21-22
%I Elsevier
%R 10.1016/j.crma.2011.10.023
%G en
%F CRMATH_2011__349_21-22_1135_0

Jung Wook Lim. The $ D+E[{\Gamma }^{⁎}]$ construction from Prüfer domains and GCD-domains. Comptes Rendus. Mathématique, Volume 349 (2011) no. 21-22, pp. 1135-1138. doi : 10.1016/j.crma.2011.10.023. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.10.023/

Bibliographie
Cité par

[1] D.D. Anderson; D.F. Anderson; M. Zafrullah Splitting the t-class group, J. Pure Appl. Algebra, Volume 74 (1991), pp. 17-37

[2] D.D. Anderson; D.F. Anderson; M. Zafrullah The ring $D + X D_{S} [X]$ and t-splitting sets, Arab. J. Sci. Eng. Sect. C Theme Issues, Volume 26 (2001), pp. 3-16

[3] D.F. Anderson; A. Ryckaert The class group of $D + M$ , J. Pure Appl. Algebra, Volume 52 (1988), pp. 199-212

[4] J.T. Arnold; J.W. Brewer On flat overrings, ideal transforms and generalized transforms of a commutative ring, J. Algebra, Volume 18 (1971), pp. 254-263

[5] H.S. Butts; C.G. Spaht Generalized quotient rings, Math. Nachr., Volume 53 (1972), pp. 181-210

[6] H.S. Butts; N. Vaughan On overrings of a domain, J. Sci. Hiroshima Univ. Ser. A-I, Volume 33 (1969), pp. 95-104

[7] G.W. Chang; B.G. Kang; J.W. Lim Prüfer v-multiplication domains and related domains of the form $D + D_{S} [Γ^{⁎}]$ , J. Algebra, Volume 323 (2010), pp. 3124-3133

[8] D. Costa; J.L. Mott; M. Zafrullah The construction $D + X D_{S} [X]$ , J. Algebra, Volume 53 (1978), pp. 423-439

[9] R. Gilmer Commutative Semigroup Rings, Univ. of Chicago Press, Chicago and London, 1984

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

[11] I. Kaplansky Commutative Rings, Polygonal Publishing House, Washington, New Jersey, 1994

[12] J.W. Lim, Generalized Krull domains and the composite semigroup ring $D + E [Γ^{⁎}]$ , submitted for publication.

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

Chain conditions in special pullbacks

Jung Wook Lim; Dong Yeol Oh

C. R. Math (2012)

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

Gyu Whan Chang

C. R. Math (2012)