Bhargava rings over subsets (Q1627640)

For an integral domain \(D\) with quotient field \(K\), its subset \(E\) and \(x\in D\) denote by \(B_x(E,D)\) the ring of all polynomials \(f\in K[X]\) with \(f(xX + e)\in D[X]\) for all \(e\in E\). In the case \(E=D\) such rings were studied by \textit{J. Yeramian} [Commun. Algebra 32, No. 8, 3043--3069 (2004; Zbl 1061.13011); J. Pure Appl. Algebra 213, No. 6, 1013--1025 (2009; Zbl 1162.13007)] and called \textit{Bhargava rings}. In the case when \(D\) is Dedekind and \(E\) is its non-empty subset, \textit{M. Bhargava} [J. Am. Math. Soc. 22, No. 4, 963--993 (2009; Zbl 1219.11047)] gave a necessary and sufficient condition for \(B_x(E,D)\) to have a regular basis, and \textit{M. Bhargava} et al. [J. Algebra 322, No. 4, 1129--1150 (2009; Zbl 1177.13051)] proved that if \(D\) is the ring of integers of an algebraic number field, then \(B_x(E,D)\) is finitely generated. In the paper under review the authors study several properties of \(B_x(E,D)\), considering in particular the behavior of \(B_x(E,D)\) under localization, its Krull and valuative dimensions and the form of its prime ideals. They show that \(B_x(E,D)\) is integrally closed if and only if \(D\) is integrally closed, and prove that it is a Bezout domain if and only if \(D\) is a field. For the entire collection see [Zbl 1392.13001].