On the number of minimal prime ideals in the completion of a local domain
Let \(\bar X\) denote the integral closure of a ring or an ideal \(X\) and \(R^*\) the completion of a local ring \(R.\) If \(R\) is a quasi-unmixed local domain with maximal ideal \(M\), \(x_ 1,...,x_ d\) a system of parameters, \(T=R[x_ 2/x_ 1,..., x_ d/x_ 1]_{M[x_ 2/x_ 1,... ,x_ d/x_ 1]}\) and \(r\) and \(k\) the numbers of maximal ideals in \(\bar R\) and \(\bar =T\) respectively, then the following inequalities hold: \(r\leq\) number of minimal prime ideals in \(R\leq\) k. One of the spinoffs is a criterion for a local domain to be analytically irreducible and another is a boundary for the number of minimal prime ideals in \(R^*\), in the non-quasi-unmixed case.
scientific article; zbMATH DE number 3993739