Integral domains with finitely many spectral semistar operations

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.