Full subsets of a noetherian ring (Q1802213)

A subset \(E\) of a domain \(R\) is called ``full'' if for each \(P\in R[X]\) such that \(P(E)\subset\mathbb{Z}\) it follows that \(P(\mathbb{Z})\subset\mathbb{Z}\). If \(R\) is a \(D\)-ring which is noetherian and 1-dimensional, such that, for each maximal ideal \({\mathfrak m}\), \(R/{\mathfrak m}\) is finite and the localization \(R_{\mathfrak m}\) is analytically irreducible, then a subset \(E\) of \(R\) is full if and only if \(E\) is dense in the \({\mathfrak m}\)-adic topology, for each maximal ideal \({\mathfrak m}\).