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Integral domains with finitely many spectral semistar operations |
Gyu Whan CHANG1,Dong Yeol OH2( ) |
1. Department of Mathematics Education, Incheon National University, Incheon 22012, Republic of Korea
2. Department of Mathematics Education, Chosun University, Gwangju 61452, Republic of Korea |
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Abstract Let D be a finite-dimensional integral domain, Spec(D) be the set of prime ideals of D, and SpSS(D) be the set of spectral semistar operations on D. Mimouni gave a complete description for the prime ideal structure of D with |SpSS(D)| = n + dim(D) for 1≤n≤5 except for the quasi-local cases of n = 4, 5. In this paper, we show that there is an integral domain D such that |SpSS(D)| = n+dim(D) for all positive integers n with n ≠2. As corollaries, we completely characterize the quasi-local domains D with |SpSS(D)| = n+dim(D) for n = 4, 5. Furthermore, we also present the lower and upper bounds of |SpSS(D)| when Spec(D) is a finite tree.
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| Keywords
(Spectral) semistar operation
prime spectrum
(Krull) dimension
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Corresponding Author(s):
Dong Yeol OH
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Issue Date: 17 November 2016
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| 1 |
Chang G W. On the cardinality of stable star operations of finite type on an integral domain. C R Math Acad Sci Paris, 2012, 350: 557–560
https://doi.org/10.1016/j.crma.2012.05.015
|
| 2 |
Finocchiaro C A, Fontana M, Spirito D. New distinguished classes of spectral spaces: a survey. arXiv: 1510.04443
|
| 3 |
Fontana M, Huckaba J. Localizing systems and semistar operations. In: Non-Noetherian Commutative Ring Theory. Math Appl, Vol 520. Dordredt: Kluwer Acad, 2000, 169–197
https://doi.org/10.1007/978-1-4757-3180-4_8
|
| 4 |
Fontana M, Loper K. Nagata rings, Kronecker function rings, and related semistar operations. Comm Algebra, 2003, 31: 4775–4805
https://doi.org/10.1081/AGB-120023132
|
| 5 |
Gilmer R. Multiplicative Ideal Theory. Queen’s Papers in Pure and Appl Math, Vol 90. Queen’s University, Kingston, Ontario, Canada, 1992
|
| 6 |
Lewis W J. The spectrum of a ring as a partially ordered set. J Algebra, 1973: 25: 419–434
https://doi.org/10.1016/0021-8693(73)90091-4
|
| 7 |
Mimouni A. Semistar-operations of finite character on integral domains. J Pure Appl Algebra, 2005, 200: 37–50
https://doi.org/10.1016/j.jpaa.2004.12.019
|
| 8 |
Mimouni A. On the prime spectrum of an integral domains with finite spectral semistar operations. Algebra Colloq, 2011, 18: 965–972
https://doi.org/10.1142/S1005386711000848
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| 9 |
Okabe A, Matsuda R. Semistar operations on integral domains. Math J Toyama Univ, 1994, 17: 1–21
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