Prime ideals lying over zero in polynomial rings (Q1895616)

Let \(R\) be a Noetherian domain with integral closure \(\overline R\) and polynomial ring \(R[x]\) in the indeterminate \(x\). Let \(P\) be a prime ideal of \(R\) and \(p\) a prime ideal of \(R[x]\) such that \(p\cap R= P\). Then \(p\) is called an integral upper to \(P\) (in \(R[x]\)) if \(p\neq PR[x]\) and \(p\) contains a monic polynomial. Let \({\mathcal P}= \{p_1,\dots, p_n\}\) be a set of integral uppers in \(R[x]\) such that each \(p\in {\mathcal P}\) is of height at least 2, and \(\mathcal Q\) be another finite set of prime ideals such that no prime ideal in \(\mathcal Q\) contains a prime ideal in \(\mathcal P\). The author summarizes the main results of the paper with the following equivalent statements: (a) either: (i) \(R\) is non-Henselian, or (ii) \(R\) is Henselian and in \(\overline R[x]\) there exist prime ideals \(\overline p_i\) lying over \(p_i\) such that \(\overline p_1+\cdots+ \overline p_n\neq \overline R[x]\); (b) there exists an integral upper \(K\) to (0) such that \(K\subseteq p\) for every \(p\in {\mathcal P}\); (c) there exists \(|R|\) integral uppers \(K\) to (0) such that \(K\subseteq p\) for every \(p\in {\mathcal P}\) but \(\overline K\not\subseteq q\) for any \(q\in {\mathcal Q}\).