On the number of semistar operations of some classes of Pr\"ufer domains
Bull. Korean Math. Soc. 2019 Vol. 56, No. 6, 1485-1495
Published online August 6, 2019
Printed November 30, 2019
Abdeslam Mimouni
King Fahd University of Petroleum \& Minerals
Abstract : The purpose of this paper is to compute the number of semistar operations of certain classes of finite dimensional Pr\"ufer domains. We prove that $|SStar(R)|=|Star(R)|+|Spec(R)|+|Idem(R)|$ where $Idem(R)$ is the set of all nonzero idempotent prime ideals of $R$ if and only if $R$ is a Pr\"ufer domain with $Y$-graph spectrum, that is, $R$ is a Pr\"ufer domain with exactly two maximal ideals $M$ and $N$ and $Spec(R)=\{(0)\subsetneq P_{1}\subsetneq\dots\subsetneq P_{n-1}\subsetneq M, N\,|\, P_{n-1}\subsetneq N\}$. We also characterize non-local Pr\"ufer domains $R$ such that $|SStar(R)|=7$, respectively $|SStar(R)|=14$.
Keywords : star operation, semistar operation, Pr\"{u}fer domain, $Y$-graph spectrum
MSC numbers : 13F05, 13A15
Supported by : The author was supported by KFUPM under research project RG 1413
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