The set of indeterminate rings of a normal pair as a partially ordered set

Abstract

Let \({R\subset S}\) be an extension of integral domains and let [R, S] be the set of intermediate rings between R and S ordered by inclusion. If (R, S) is normal pair and [R, S] is finite, we do prove that there exists a semi-local Prufer ring T with quotient field K such that \({[R,S]\cong \lbrack T,K]}\) (as a partially ordered set). Consequently, any problem relative to the finiteness conditions in [R, S] can be investigated in the particular case where R is a semi-local Prufer ring with quotient field S.