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

Abstract


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 $|Spec(D)| +1 =  |SFc(D)|$.

Keywords


semistar operation of finite character, Pr\"ufer domain

Subject classification

3A15, 13G05, 13F05.

Sponsor(s)

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

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References


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