On the dimension theory of overrings of an integral domain

Abstract

Let D be an integral domain with identity which has quotient field L. If there exists a chain P c Pi c... * P%, of n + 1 prime ideals of D, where Pn, C D, but no such chain of n +2 prime ideals, then we say that D has dimension n and we write dim D = n [6]. In [6] and [7] Seidenberg has shown that if dim D = n, and if D is a Noetherian domain or a Priufer domain, then dim D[X1, .. ., Xm]=n+m, where X1, .. ., Xm are indeterminates over D. In the special case in which dim D= 1 he has proved that the following statements are equivalent. (1) dim D[X1] =2. (2) dim D[Xi, ..., Xm]=m+ 1 for any m. More recently Gilmer has established the equivalence of the following properties for an n-dimensional domain D [1]. (3) Every domain between D and L has dimension less than or equal to n. (4) dim D[tj, ... ., t2j n for {ti, . ., tj(-L. For n = 1 he further showed that (3) and (4) are equivalent to (1). In this paper we consider domains D having finite dimension n and having the property that each domain between D and its quotient field has dimension less than or equal to c for some positive integer c ? n. For such a domain we obtain equivalent statements analogous to statements (1)-(4). The main results of this paper are contained in Theorems 2 and 5. Throughout this paper D will denote an integral domain with identity having quotient field L, and X, X1,..., Xm will denote indeterminates over D. By an overring of D we mean an integral domain D' such that D D' ' L. By a valuation overring of D we mean an overring of D which is a valuation ring. Our notation will be that of Zariski-Samuel [8] with the one exception: c denotes proper containment and c denotes containment.