Note on overrings without a specified element

Abstract

Anderson-Dobbs-Huckaba ([ADH]) showed that, if each s-overring of D is a PVD, then each overring of D is seminormal. Also they state that the converse does not hold ([ADH, Remark 3.3]). They constructed a domain D such that each overring of D is seminormal, and has an s-overring which is not a PVD. But it does not seem that they constructed an s-overring T of D concretely which is not a PVD. In this paper, we will answer to these Questions. Further, we will give more definite conditions when D or S is integrally closed or 1-dimensional or Noetherian. Next, we will supplement [ADH, Example 3.2] to give a domain D and an s-overring T of D such that each overring of D is seminormal, and T is not a PVD. We note that [ADH] holds for any g-monoid S ([KM], [MK1] and [MK2]). If, for each maximal ideal M of D, the integral closure of DM is a valuation ring, then D is called an i-domain ([P]). The integral closure of D is denoted by D', and the integral closure of S is denoted by S'. If S' is a valuation seinigroup, then S is called an i-seinigroup. The following is a semigroup version of [ADH, Proposition 3.1].