Abstract
If T is a (commutative unital) ring extension of a ring R, then Λ(T /R) is defined to be the supremum of the lengths of chains of intermediate fields between R P /P R P and T Q /QT Q , where Q varies over Spec(T) and P:= Q ∩ R. The invariant σ(R):= sup Λ(T/R), where T varies over all the overrings of R. It is proved that if Λ(S/R)< ∞ for all rings S between R and T, then (R, T) is an INC-pair; and that if (R, T) is an INC-pair such that T is a finite-type R-algebra, then Λ(T/R)< ∞. Consequently, if R is a domain with σ(R) < ∞, then the integral closure of R is a Prüfer domain; and if R is a Noetherian G-domain, then σ(R) < ∞, with examples showing that σ(R) can be any given non-negative integer. Other examples include that of a onedimensional Noetherian locally pseudo-valuation domain R with σ(R)=∞.
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Dobbs, D.E. On the finiteness of a field-theoretic invariant for commutative rings. Rend. Circ. Mat. Palermo 58, 327–336 (2009). https://doi.org/10.1007/s12215-009-0027-x
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DOI: https://doi.org/10.1007/s12215-009-0027-x
Keywords
- Commutative ring
- Overring
- Prime ideal
- Intermediate field
- Finite-type algebra
- INC-pair
- Integral domain
- Prüfer domain
- Integral closure
- G-domain
- Pseudo-valuation domain
- D+M construction