Comaximizing and splitting primes (Q2277029)

Let R be a ring, let P be a prime ideal of R, and let B be a finite integral extension ring of R. Then it is said P \(n-co\max imally\) splits in B in case there exist at least n pairwise comaximal prime ideals of B that lie over P. With this terminology, the following are the two main results in this paper: (1) Let \(n\geq 2\) be a positive integer and let C be a collection of prime ideals of a non-Henselian Noetherian domain R such that \(\cap \{P;P\in C\}\neq (0).\) Then there exists a finite integral extension domain \(A_ n\) of R in which every P in C n-comaximally splits. (2) Let \(n\geq 2\) be a positive integer and let \(P_ 1,...,P_ g\) (g\(\geq 2)\) be (not necessarily distinct) nonzero prime ideals in a non-Henselian semi-local domain R. Then there exists a simple integral extension domain \(R[e_ n]\) of R that has ng pairwise comaximal prime ideals \(Q_{i,j}\) such that \(Q_{i,j}\cap R=P_ i\) for \(i=1,...,g\) and \(j=1,...,n\).