An answer to a problem about the number of overrings

摘要

Let R be an integral domain with only finitely many overrings, equivalently, a domain such that its integral closure R' is a Prufer domain with finite spectrum and there are only finitely many rings between R and R'. Jaballah solved the problem of counting the overrings in the case R = R' but left the general case as an open problem [A. Jaballah, The number of overrings of an integrally closed domain, Expo. Math. 23 (2005) 353-360, Problem 3.4]. The purpose of this paper is to provide a solution to that problem.