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Some more combinatorics results on Nagata extensions. (English) Zbl 1346.13012

Summary: We show that the length of a ring extension \(R \subseteq S\) is preserved under the formation of the Nagata extension \(R(X) \subseteq S(X)\). A companion result holds for the Dobbs-Mullins invariant. D. Dobbs and the authors proved elsewhere that the cardinal number of the set \([R, S]\) of subextensions of \(R \subseteq S\) is preserved under the formation of Nagata extension when \(|[R(X),S(X)]|\) is finite. We show that in the only pathological case, namely \(R \subseteq S\) is subintegral, then \(|[R,S]|\) is preserved if and only if it is either infinite or finite and \(R \subseteq S\) is arithmetic; that is, \([R,S]\) is locally a chain. The last section gives properties of arithmetic extensions and their links with Prüfer extensions.

MSC:

13B02 Extension theory of commutative rings
13B21 Integral dependence in commutative rings; going up, going down
13B25 Polynomials over commutative rings
12F05 Algebraic field extensions
13B22 Integral closure of commutative rings and ideals
13B30 Rings of fractions and localization for commutative rings
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