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

Abstract

Let R ⊂ 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 Prüfer ring T with quotient field K such that [R, S] ∼ = [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 Prüfer ring with quotient field S.

Maximal non valuation domains in an integral domain

“…Now, by [12], Lemma 6.2, (R, S) is a normal pair. Therefore, by [3] Remark 2.3. As we have already seen that if R is integrally closed in S, R ⊂ S, and dim(R, S) is finite, then the conditions that the pair (R, S) is normal and R is local are not necessary for |[R, S]| = 1 + dim(R, S), however these are sufficient.…”

“…However, R is not local. In the next theorem, we prove that there is a complete class of ring extensions which counters [3], Proposition 6 (i).…”

“…Therefore, T is a valuation domain by [19], Theorem 7 and hence T is a λ-domain by [14], Corollary 1.5. Thus, R ⊂ S is a λ-extension and Supp(S/R) is finite with a unique maximal element by [3], Theorem 4.…”

“…In [3], Ayache introduced the notion of dim(R, S) as the number of terms of the longest maximal chains in Supp(S/R). Ayache, in [3], Proposition 6 (i), showed the following: If R is integrally closed in S and dim(R, S) is finite, then (R, S) is a normal pair and R is local if and only if |[R, S]| = 1 + dim(R, S).…”

“…Then by [2], Theorem 9, (R, S) is a normal pair and Supp(S/R) is finite. Now, by [3], Theorem 4, there exists a semi local Prüfer domain T such that |[T, qf (T )]| = 1 + dim(T ). Therefore, T is a valuation domain by [19], Theorem 7 and hence T is a λ-domain by [14], Corollary 1.5.…”