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http://dx.doi.org/10.4134/BKMS.b181216

ON THE NUMBER OF SEMISTAR OPERATIONS OF SOME CLASSES OF PRUFER DOMAINS  

Mimouni, Abdeslam (Department of Mathematics and Statistics King Fahd University of Petroleum & Minerals)
Publication Information
Bulletin of the Korean Mathematical Society / v.56, no.6, 2019 , pp. 1485-1495 More about this Journal
Abstract
The purpose of this paper is to compute the number of semistar operations of certain classes of finite dimensional $Pr{\ddot{u}}fer$ domains. We prove that ${\mid}SStar(R){\mid}={\mid}Star(R){\mid}+{\mid}Spec(R){\mid}+ {\mid}Idem(R){\mid}$ where Idem(R) is the set of all nonzero idempotent prime ideals of R if and only if R is a $Pr{\ddot{u}}fer$ domain with Y -graph spectrum, that is, R is a $Pr{\ddot{u}}fer$ domain with exactly two maximal ideals M and N and $Spec(R)=\{(0){\varsubsetneq}P_1{\varsubsetneq}{\cdots}{\varsubsetneq}P_{n-1}{\varsubsetneq}M,N{\mid}P_{n-1}{\varsubsetneq}N\}$. We also characterize non-local $Pr{\ddot{u}}fer$ domains R such that ${\mid}SStar(R){\mid}=7$, respectively ${\mid}SStar(R){\mid}=14$.
Keywords
star operation; semistar operation; $Pr{\ddot{u}}fer$ domain; Y-graph spectrum;
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