Star operations related to polynomial closure
Let \(D\) be an integral domain with quotient field \(K\). Recall that a fractional ideal \(I\) of \(D\) is a \(D\)-submodule of \(K\) such that \(dI\subseteq D\) for some \(0\not=d\in D\). We denote by \(\mathcal{F}(D)\) the set of fractional ideals of \(D\). A star operation on \(D\) is a map \(*:\mathcal{F}(D)\longrightarrow\mathcal{F}(D)\), \(I\longrightarrow I^*\), such that for each \(I,J\in\mathcal{F}(D)\), \(d\in K\), \(I\subseteq I^*\); if \(I\subseteq J\), then \(I^*\subseteq J^*\); \((I^*)^*=I^*\); \((dI)^*=dI^*\); \(D^*=D\).\N\NA typical example of star-operation is the \(v\)-operation (or divisorial closure) \(I_v=D:(D:I)\) where \((J:L)=\{x\in K;xL\subseteq J\}\).\N\NLet \(E\) be a subset of \(K\). Then \(\mathrm{Int}(E,D)=\{f(X)\in K[X]; f(E)\subseteq D\}\), is a subring of \(K[X]\), called the ring of integer-valued polynomials over \(D\). The polynomial closure of \(E\) in \(D\) is the largest subset \(F\) of \(K\) such that \(\mathrm{Int}(E,D)=\mathrm{Int}(F,D)\). It is well known that polynomial closure is a star operation in any domain \(D\).\N\NIn the paper under review, the authors show that the polynomial closure and the \(v\)-operation coincide for two wide classes of domains, namely the integrally closed domains and the domains of residue characteristic \(0\). The case of characteristic \(p\) is also investigated.
scientific article; zbMATH DE number 7903227