Minimal Zero-Dimensional Extensions of Rings of Dimension Greater than One

Abstract

Let R be a commutative ring with Noetherian spectrum in which zero is a primary ideal. We determine the minimal zero-dimensional extensions of R when every regular prime ideal of R is contained in only finitely many prime ideals. This extends previous results of the first author for dim R ≤ 1. We also present a characterization of the partially ordered set of prime ideals in a ring with Noetherian spectrum.