Frontiers of Mathematics in China >

2017 , Vol. 12 >Issue 1: 35 - 49

DOI: https://doi.org/10.1007/s11464-016-0587-y

RESEARCH ARTICLE

Integral domains with finitely many spectral semistar operations

  • Gyu Whan CHANG 1 ,
  • Dong Yeol OH , 2
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  • 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

Received date: 16 Nov 2015

Accepted date: 18 Apr 2016

Published date: 01 Feb 2017

Copyright

2016 Higher Education Press and Springer-Verlag Berlin Heidelberg
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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.

Key words： (Spectral) semistar operation; prime spectrum; (Krull) dimension

Cite this article

Gyu Whan CHANG , Dong Yeol OH . Integral domains with finitely many spectral semistar operations[J]. Frontiers of Mathematics in China, 2017 , 12(1) : 35 -49 . DOI: 10.1007/s11464-016-0587-y

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