The Skolem closure as a semistar operation (Q2199720)

Let \(D\) be a commutative domain with quotient field \(K\). denote by \(Int(D)\) the ring of polynomials \(f\in K[X]\) with \(f(D)\subset D\) and for \(d\in D\) and an ideal \(I\) of \(Int(D)\) put \[I(d)=\{f(d):\ f\in I\}.\] The Skolem closure \(I^*\) of an ideal \(I\) of \(Int(D)\) is defined as the largest ideal satisfying \(I^*(d)=I(d)\) for all \(d\in D\), and \(D\) has the Skolem property if for every proper finitely generated ideal \(I\) there is \(d\in D\) such that \(I(d)\) is a proper ideal [\textit{Th. Skolem}, Norske Vid. Selsk. Forhdl. 9, 111--113 (1936; Zbl 0016.24504)]. In Sect. 2 these notions are extended to \(A\)-submodules of \(K(x)\), where \(A\) is a domain satisfying \(D[X]\subset A\subset K(x)\). Star operations (as defined by \textit{W. Krull} [Math. Z. 41, 545--577 (1936; Zbl 0015.00203)]) are used in Sect. 3 to describe ideals in \(Int(Z)\) with the Skolem and strong Skolem properties, and Sect. 4 is devoted to the study of relations between the generalization of Skolem closure given in Sect. 2 and semistar operations, which differ from star operations by omission of the condition \(A^*=A\).