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http://dx.doi.org/10.4134/BKMS.b170233

ON GENERALIZED KRULL POWER SERIES RINGS

Le, Thi Ngoc Giau (Faculty of Mathematics and Statistics Ton Duc Thang University)
Phan, Thanh Toan (Faculty of Mathematics and Statistics Ton Duc Thang University)

Publication Information

Bulletin of the Korean Mathematical Society / v.55, no.4, 2018 , pp. 1007-1012 More about this Journal

Abstract

Let R be an integral domain. We prove that the power series ring R[[X]] is a Krull domain if and only if R[[X]] is a generalized Krull domain and t-dim $R{\leq}1$ , which improves a well-known result of Paran and Temkin. As a consequence we show that one of the following statements holds: (1) the concepts "Krull domain" and "generalized Krull domain" are the same in power series rings, (2) there exists a non-t-SFT domain R with t-dim R > 1 such that t-dim R[[X]] = 1.

Keywords

generalized Krull domain; Krull domain; power series ring;

Citations & Related Records

Reference

1	J. T. Arnold, Krull dimension in power series rings, Trans. Amer. Math. Soc. 177 (1973), 299-304. DOI
2	G. W. Chang, B. G. Kang, and P. T. Toan, The Krull dimension of power series rings over almost Dedekind domains, J. Algebra 438 (2015), 170-187. DOI
3	G. W. Chang and D. Y. Oh, When D((X)) and D{{X}} are Prufer domains, J. Pure Appl. Algebra 216 (2012), no. 2, 276-279. DOI
4	M. D. Fried and M. Jarden, Field Arithmetic, third edition, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 11, Springer-Verlag, Berlin, 2008.
5	R. Gilmer, Power series rings over a Krull domain, Pacific J. Math. 29 (1969), 543-549. DOI
6	R. Gilmer, Multiplicative Ideal Theory, Marcel Dekker, Inc., New York, 1972.
7	M. Griffin, Some results on v-multiplication rings, Canad. J. Math. 19 (1967), 710-722. DOI
8	B. G. Kang, Prufer v-multiplication domains and the ring $R[X]_{N_v}$ , J. Algebra 123 (1989), no. 1, 151-170. DOI
9	B. G. Kang, K. A. Loper, T. G. Lucas, M. H. Park, and P. T. Toan, The Krull dimension of power series rings over non-SFT rings, J. Pure Appl. Algebra 217 (2013), no. 2, 254- 258. DOI
10	B. G. Kang and M. H. Park, A localization of a power series ring over a valuation domain, J. Pure Appl. Algebra 140 (1999), no. 2, 107-124. DOI
11	B. G. Kang and M. H. Park, A note on t-SFT-rings, Comm. Algebra 34 (2006), no. 9, 3153-3165. DOI
12	E. Paran, Split embedding problems over complete domains, Ann. of Math. (2) 170 (2009), no. 2, 899-914. DOI
13	B. G. Kang and M. H. Park, Krull-dimension of the power series ring over a nondiscrete valuation domain is uncountable, J. Algebra 378 (2013), 12-21. DOI
14	K. A. Loper and T. G. Lucas, Constructing chains of primes in power series rings, J. Algebra 334 (2011), 175-194. DOI
15	K. A. Loper and T. G. Lucas, Constructing chains of primes in power series rings, II, J. Algebra Appl. 12 (2013), no. 1, 1250123, 30 pp.
16	R. Weissauer, Der Hilbertsche Irreduzibilitatssatz, J. Reine Angew. Math. 334 (1982), 203-220.
17	E. Paran and M. Temkin, Power series over generalized Krull domains, J. Algebra 323 (2010), no. 2, 546-550. DOI
18	F. Pop, Henselian implies large, Ann. of Math. (2) 172 (2010), no. 3, 2183-2195. DOI