On the complete integral closure of an integral domain

Abstract

We consider in this paper only commutative rings with identity. When R is considered as a subring of S it will always be assumed that R and S have the same identity. If R is a subring of S an element s of S said to be integral over R if s is the root of a monic polynomial with coefficients in R. Following Krull [8], p. 102, we say s is almost integral over R provided all powers of s belong to a finite R-submodule of S. If R1 is the set of elements of S almost integral over R we say R1 is the complete integral closure of R in S.