Geodesic
Power Series Rings Over Prüfer v-multiplication Domains. II
Canadian mathematical bulletin, Tome 60 (2017) no. 1, pp. 63-76

Voir la notice de l'article provenant de la source Cambridge University Press

Let $D$ be an integral domain, ${{X}^{1}}\left( D \right)$ be the set of height-one prime ideals of $D$ , $\left\{ {{X}_{\beta }} \right\}$ and $\left\{ {{X}_{\alpha }} \right\}$ be two disjoint nonempty sets of indeterminates over $D$ , $D\left[ \left\{ {{X}_{\beta }} \right\} \right]$ be the polynomial ring over $D$ , and $D\left[ \left\{ {{X}_{\beta }} \right\} \right]{{\left[\!\left[ \left\{ {{X}_{\alpha }} \right\} \right]\!\right]}_{1}}$ be the first type power series ring over $D\left[ \left\{ {{X}_{\beta }} \right\} \right]$ . Assume that $D$ is a Prüfer $v$ -multiplication domain $\left( \text{P}v\text{MD} \right)$ in which each proper integral $t$ -ideal has only finitely many minimal prime ideals (e.g., $t$ - $\text{SFT}$ $\text{P}v\text{MDs}$ , valuation domains, rings of Krull type). Among other things, we show that if ${{X}^{1}}\left( D \right)\,=\,\phi$ or ${{D}_{p}}$ is a $\text{DVR}$ for all $P\,\in \,{{X}^{1}}\left( D \right)$ , then $D\left[ \left\{ {{X}_{\beta }} \right\} \right]{{\left[\!\left[ \left\{ {{X}_{\alpha }} \right\} \right]\!\right]}_{1D-\left\{ 0 \right\}}}$ is a Krull domain. We also prove that if $D$ is a $t$ - $\text{SFT}\text{P}v\text{MD}$ , then the complete integral closure of $D$ is a Krull domain and $\text{ht}\left( M\left[ \left\{ {{X}_{\beta }} \right\} \right]{{\left[\!\left[ \left\{ {{X}_{\alpha }} \right\} \right]\!\right]}_{1}} \right)\,=\,1$ for every height-one maximal $t$ -ideal $M$ of $D$ .
DOI : 10.4153/CMB-2016-046-5
Mots-clés : 13A15, 13F05, 13F25, Krull domain, PvMD, multiplicatively closed set of ideals, power series ring 
Chang, Gyu Whan. Power Series Rings Over Prüfer v-multiplication Domains. II. Canadian mathematical bulletin, Tome 60 (2017) no. 1, pp. 63-76. doi: 10.4153/CMB-2016-046-5
@article{10_4153_CMB_2016_046_5,
     author = {Chang, Gyu Whan},
     title = {Power {Series} {Rings} {Over} {Pr\"ufer} v-multiplication {Domains.} {II}},
     journal = {Canadian mathematical bulletin},
     pages = {63--76},
     year = {2017},
     volume = {60},
     number = {1},
     doi = {10.4153/CMB-2016-046-5},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2016-046-5/}
}
TY  - JOUR
AU  - Chang, Gyu Whan
TI  - Power Series Rings Over Prüfer v-multiplication Domains. II
JO  - Canadian mathematical bulletin
PY  - 2017
SP  - 63
EP  - 76
VL  - 60
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-2016-046-5/
DO  - 10.4153/CMB-2016-046-5
ID  - 10_4153_CMB_2016_046_5
ER  - 
%0 Journal Article
%A Chang, Gyu Whan
%T Power Series Rings Over Prüfer v-multiplication Domains. II
%J Canadian mathematical bulletin
%D 2017
%P 63-76
%V 60
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-2016-046-5/
%R 10.4153/CMB-2016-046-5
%F 10_4153_CMB_2016_046_5

[1] [1] Anderson, D. D., D. E Anderson, and Zafrullah, M., The ring D + XDg [X] and t-splitting sets. In: Commutative algebra, Arab J. Sci. Eng. Sect. C Theme Issues 26(2001), 3–16. Google Scholar

[2] [2] Anderson, D. D., Chang, G. W., and Zafrullah, M., Integral domains of finite t-character. J. Algebra 396(2013), 169–183. http://dx.doi.Org/10.1016/j.jalgebra.2013.08.014 Google Scholar

[3] [3] Anderson, D. D., Kang, B. G., and Park, M. H., Anti-Archimedean rings and power series rings. Comm. Algebra 26(1998), no. 10 ,3223-3238. http://dx.doi.Org/10.1080/00927879808826338 Google Scholar

[4] [4] Arnold, J. T., Power series rings over Prtifer domains. Pacific J. Math. 44(1973), 1–11. http://dx.doi.Org/10.2140/pjm.1973.44.1 Google Scholar

[5] [5] Arnold, J. T., Power series rings with finite Krull dimension. Indiana Univ. Math. J. 31(1982), no. 6, 897–911. http://dx.doi.Org/10.1512/iumj.1982.31.31061 Google Scholar

[6] [6] Arnold, J. T. and Brewer, J. W., On flat overrings, ideal transforms and generalized transforms of a commutative ring. J. Algebra 18(1971), 254–263. http://dx.doi.Org/10.1016/0021-8693(71)9OO58-5 Google Scholar

[7] [7] Chang, G. W., Spectral localizing systems that are t-splitting multiplicative sets of ideals. J. Korean Math. Soc. 44(2007), 863–872. http://dx.doi.Org/10.4134/JKMS.2007.44.4.863 Google Scholar

[8] [8] Chang, G. W., Power series rings over Priifer v-multiplication domains. J. Korean Math. Soc. 53(2016), no. 2, 447–459. http://dx.doi.Org/10.4134/JKMS.2016.53.2.447 Google Scholar

[9] [9] Chang, G. W. and Oh, D. Y., The rings D((X))i and D﹛﹛X﹜﹜;. J. Algebra Appl. 12(2013), no. 2, 1250147. http://dx.doi.Org/10.1142/S0219498812 501472 Google Scholar

[10] [10] Decruyenaere, E. and Jespers, E., Priifer domains and graded rings. J. Algebra 150(1992), no. 2, 308–320. http://dx.doi.Org/10.1016/S0021-8693(05)80034-1 Google Scholar

[11] [11] Dobbs, D., Houston, E., Lucas, T., and Zafrullah, M., t-linked overrings and Priifer v-multiplication domains. Comm. Algebra 17(1989), no. 11, 2835–2852. http://dx.doi.org/10.1080/0092787890882387. Google Scholar

[12] [12] El Baghdadi, S., On a class of Priifer v-multiplication domains. Comm. Algebra 30(2002), no. 8, 3723-3724. http://dx.doi.Org/10.1081/ACB-120005815 Google Scholar

[13] [13] El Baghdadi, S. and Kim, H., Generalized Krull semigroup rings. Comm. Algebra 44(2016), no. 4, 1783–1794. http://dx.doi.Org/10.1080/00927872.2015.1027378 Google Scholar

[14] [14] Fontana, M., Gabelli, S., and Houston, E., UMT-domains and domains with Priifer integral closure. Comm. Algebra 26(1998), 1017–1039. http://dx.doi.Org/10.1080/00927879808826181 Google Scholar

[15] [15] Fontana, M., Huckaba, J. A., and Papick, I. J., Priifer domains. Monographs and Textbooks in Pure and Applied Math., 203, Marcel Dekker, New York, 1997. Google Scholar

[16] [16] Fossum, R. M., The divisor class group of a Krull domain. Springer-Verlag, New York-Heidelberg, 1973. Google Scholar

[17] [17] Gabelli, S., Generalized Dedekind domains. In: Multiplicative ideal theory in commutative algebra, Springer, New York, 2006, pp. 189–206. http://dx.doi.Org/10.1007/978-0-387-36717-0J2 Google Scholar

[18] [18] Gilmer, R., Power series rings over a Krull domain. Pacific J. Math. 29(1969), 543–549. http://dx.doi.Org/10.2140/pjm.1969.29.543 Google Scholar

[19] [19] Gilmer, R., Multiplicative ideal theory. Pure and Applied Mathematics, 12, Marcel Dekker, New York, 1972. Google Scholar

[20] [20] Griffin, M., Some results on v-multiplication rings. Cana Math d. J. 19(1967), 710–722. http://dx.doi.Org/10.4153/CJM-1967-065-8 Google Scholar

[21] [21] Griffin, M., Rings ofKrull type. J. Reine Angew. Math. 229(1968), 1–27. Google Scholar

[22] [22] Kang, B. G., Prüfer v-multiplication domains and the ring R[X]. J. Algebra 123(1989), no. 1, 151–170. http://dx.doi.Org/10.1 01 6/0021-8693(89)90040-9 Google Scholar

[23] [23] Kang, B. G. and Park, M. H., On Mockor's question. J. Algebra 216(1999), 481–510. http://dx.doi.Org/10.1006/jabr.1 998.7785 Google Scholar

[24] [24] Kang, B. G., A note on t-SFT-rings. Comm. Algebra 34(2006), no. 9, 3153–3165. http://dx.doi.Org/10.1080/00927870600639476 Google Scholar

[25] [25] Mott, J. L., On the complete integral closure of an integral domain ofKrull type. Math. Ann. 173(1967), 238–240. http://dx.doi.Org/10.1007/BF01361714 Google Scholar

[26] [26] Mott, J. L. and Zafrullah, M., On Priifer v-multiplication domains. Manuscripta Math. 35(1981), 1–26. http://dx.doi.Org/10.1007/BF011 68446 Google Scholar

[27] [27] Ohm, J., Some counterexamples related to integral closure in D[[X]]. Trans. Amer. Math. Soc. 122(1966), 321–333. Google Scholar

[28] [28] Paran, E. and Temkin, M., Power series over generalized Krull domains. J. Algebra 323(2010), no. 2, 546–550. http://dx.doi.Org/10.1016/j.jalgebra.2009.08.011 Google Scholar

[29] [29] Popescu, N., On a class of Priifer domains. Rev. Roumaine Math. Pures Appl. 29(1984), 777–786. Google Scholar

[30] [30] Samuel, P., On unique factorization domains. Illinois J. Math. 5(1961), 1–17. Google Scholar

Cité par Sources :