Picavet, Gabriel; Picavet-L’Hermitte, Martine Some more combinatorics results on Nagata extensions. (English) Zbl 1346.13012 Palest. J. Math. 5, Spec. Iss., 49-62 (2016). 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. Cited in 1 ReviewCited in 12 Documents 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 Keywords:arithmetic extension; FIP; FCP; FMC extension; minimal extension; Prüfer extension; integral extension; support of a module; Nagata ring; t-closure PDFBibTeX XMLCite \textit{G. Picavet} and \textit{M. Picavet-L'Hermitte}, Palest. J. Math. 5, 49--62 (2016; Zbl 1346.13012) Full Text: Link