On Complete Integral Closure and Archimedean Valuation Domains

Suppose D is an integral domain with quotient eld K and that L is an extension eld of K. We show in Theorem 4 that if the complete integral closure of D is an intersection of Archimedean valuation domains on K, then the complete integral closure of D in L is an intersection of Archimedean valuation domains on L; this answers a question raised by All rings considered in this paper are assumed to be commutative and to contain a unity element. If R is a subring of S, we assume that the unity element of S belongs to R, and hence is the unity element of R. If R is a subring of S and s 2 S, then as is well-known, s is integral over R if and only if Rs] is a nitely generated R-module. On the other hand, if Rs] is contained in a nitely generated R-submodule of S, then s is said to be almost integral over R. Clearly s is almost integral over R if it is integral over R; the converse holds if R is Noetherian, but not in general. We denote by C(R; S) the set of elements of S that are almost integral over R; C(R; S) is a subring of S containing R, and is called the complete integral closure of R in S. If C(R; S) = S, we say that S is almost integral over R, and on the opposite extreme where C(R; S) = R, we say that R is completely integrally closed in S. If S is the total quotient ring of R, then C(R; S) is the complete integral closure of R, and if C(R; S) = R, then R is said to be completely integrally closed. The concept of almost integrality was rst considered by Krull 1 in his famous 1932 paper Allgemeine Bewertungstheorie 5], in which he introduced the notion of a general valuation and proved, among other things, that an integral domain is integrally closed if and only if it is an intersection of valuation domains 5, Satz 7]. Krull also proved 5, Satz 8] that a valuation domain V is completely 1 More precisely, Krull considers the concept in the case where R is an integral domain and S is the quotient eld of R. In this case it is easy to show that s 2 S is almost integral over …