no. 21-22
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

Lim, Jung Wook1 
1 Department of Mathematics, Pohang University of Science and Technology, Pohang 790-784, Republic of Korea
Comptes Rendus. Mathématique, Tome 349 (2011) no. 21-22, pp. 1135-1138.

Soit D⊊E une extension dʼanneaux commutatifs intègres, Γ un monoïde commutatif simplifiable sans torsion non trivial tel que Γ∩−Γ={0}. On note Γ⁎=Γ∖{0} et soit D+E[Γ⁎]={f∈E[Γ]|f(0)∈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∩−Γ={0}, Γ⁎=Γ∖{0} and D+E[Γ⁎]={f∈E[Γ]|f(0)∈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 :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2011.10.023
Lim, Jung Wook 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 = {Lim, Jung Wook},
     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},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2011.10.023/}
}
TY  - JOUR
AU  - Lim, Jung Wook
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
UR  - http://www.numdam.org/articles/10.1016/j.crma.2011.10.023/
DO  - 10.1016/j.crma.2011.10.023
LA  - en
ID  - CRMATH_2011__349_21-22_1135_0
ER  - 
%0 Journal Article
%A Lim, Jung Wook
%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
%U http://www.numdam.org/articles/10.1016/j.crma.2011.10.023/
%R 10.1016/j.crma.2011.10.023
%G en
%F CRMATH_2011__349_21-22_1135_0
Lim, Jung Wook. The $ D+E[{\Gamma }^{⁎}]$ construction from Prüfer domains and GCD-domains. Comptes Rendus. Mathématique, Tome 349 (2011) no. 21-22, pp. 1135-1138. doi : 10.1016/j.crma.2011.10.023. http://www.numdam.org/articles/10.1016/j.crma.2011.10.023/

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

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

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

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

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

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

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

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

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

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

[11] Kaplansky, I. 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 :