Polynomial closure in essential domains
Let \(D\) be an integral domain with quotient field \(K\). For each subset \(E\subseteq K\), \(\text{Int}(E,D)=\{f\in K[X]; f(E)\subseteq D\}\) is called the ring of \(D-\)integer-valued polynomials over \(E\). We set \(\text{Int}(D)=\text{Int}(D,D)\). The polynomial closure of \(E\) is defined as the set \(\text{cl}_D(E)=\{x\in K; f(x)\in D,\forall f\in \text{Int}(E,D)\}\), that is, \(\text{cl}_D(E)\) is the largest subset \(F\subseteq K\) such that \(\text{Int}(E,D)=\text{Int}(F,D)\). The \(v\)-closure (or, the divisorial closure) of a fractional ideal \(I\) of \(D\) is \(I_v=(I^{-1})^{-1}\) where \(I^{-1}=\{x\in K; xI\subseteq D\}\). In section 1, the authors introduce a new star operation \(\text{Cl}(-)\) in the following way. For any nonzero fractional ideal \(I\) of \(D\), \(\text{Cl}(I)=\{x\in K; f(x)\in D,\forall f\in D[{X\over I}]\}\) where \(D[{X\over I}]= \bigcap_{a\in I\setminus (0)} [{X\over a}]\). The main result is that \(\text{cl}_D(I)\subseteq \text{Cl}_D(I)\subseteq I_v\). The first inclusion becomes equality when \(\text{Int}(D)=D[X]\). Recall that an essential domain is a domain \(D\) such that \(D=\bigcap_{P\in{\mathcal P}}D_P\) (*) where \({\mathcal P}\subseteq\text{spec}(D)\) and each \(D_P\) is a valuation domain. A domain \(D\) is Krull-type if it is essential and the intersection (*) is locally finite. In section 2, the authors prove that \(\text{Cl}(-)\) is equal to the divisorial closure in any essential domain. They also show that the divisorial closure and the polynomial closure of ideals are the same in a Krull-type domain.
scientific article; zbMATH DE number 2175533