Korean J. Math.  Vol 22, No 3 (2014)  pp.455-462
DOI: https://doi.org/10.11568/kjm.2014.22.3.455

On the cardinality of semistar operations of finite character on integral domains

Gyu Whan Chang


Let $D$ be an integral domain with $Spec(D)$ finite, $K$ the quotient field of $D$, $[D,K]$ the set of rings between $D$ and $K$, and $SFc(D)$ the set of semistar operations of finite character on $D$. It is well known that $|Spec(D)| \leq |SFc(D)|$. In this paper, we prove that $|Spec(D)| = |SFc(D)|$ if and only if $D$ is a valuation domain, if and only if $|Spec(D)| = |[D,K]|$. We also study integral domains $D$ such that $|Spec(D)| +1 =  |SFc(D)|$.


semistar operation of finite character, Pr\"ufer domain

Subject classification

3A15, 13G05, 13F05.


This work was supported by the Incheon National University Research Fund in 2013 (Grant No. 20130395).

Full Text:



M. Fontana and J. Huckaba, in: S. Chapman, S. Glaz (Eds.), Localizing systems and semistar operations, Non-Noetherian Commutative Ring Theory, Vol. 520, Kluwer Academic Publisher, Dordecht, 2000, pp. 169–197. (Google Scholar)

M. Fontana and K.A. Loper, Nagata rings, Kronecker function rings and related semistar operations, Comm. Algebra 31 (2003), 4775–4805. (Google Scholar)

R. Gilmer, Multiplicative Ideal Theory, Marcel Dekker, New York, 1972. (Google Scholar)

E. Houston and M. Zafrullah, On t-invertibility II, Comm. Algebra 17 (1989),1955–1969. (Google Scholar)

B.G. Kang, Pru ̈fer v-multiplication domains and the ring R[X]Nv , J. Algebra 123 (1989), 151–170. (Google Scholar)

R. Matsuda, Note on valuation rings and semistar operations, Comm. Algebra 28 (2000), 2515–1519. (Google Scholar)

R. Matsuda, Note on the number of semistar operations III, In: A. Badawi, ed. Commutative Rings, 2002, Nova Science Publisher, pp. 77–81. (Google Scholar)

A. Mimouni, Semistar-operations of finite character on integral domains, J. Pure Appl. Algebra 200 (2005), 37–50. (Google Scholar)

A. Mimouni and M. Samman, Semistar operations on valuation domains, Internat. J. Commutative Rings 2 (2003), 131–141. (Google Scholar)

A. Okabe and R. Matusada, Semistar operations on integral domains, Math. J. Toyama Univ. 17 (1994), 1–21. (Google Scholar)

G. Picozza, Star operations on overrings and semistar operations, Comm. Algebra 33 (2005), 2051–2073. (Google Scholar)


  • There are currently no refbacks.

ISSN: 1976-8605 (Print), 2288-1433 (Online)

Copyright(c) 2013 By The Kangwon-Kyungki Mathematical Society, Department of Mathematics, Kangwon National University Chuncheon 21341, Korea Fax: +82-33-259-5662 E-mail: kkms@kangwon.ac.kr