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On the Integral Degree of Integral Ring Extensions

Part of: Ring extensions and related topics Field extensions Integral domains

Published online by Cambridge University Press:  20 August 2018

José M. Giral ,
Liam O'Carroll ,
Francesc Planas-Vilanova  and
Bernat Plans
José M. Giral*
Affiliation:
Departament d’Àlgebra i Geometria, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, E-08007 Barcelona, Spain (giral@ub.edu)
Liam O'Carroll
Affiliation:
Maxwell Institute for Mathematical Sciences, School of Mathematics, University of Edinburgh, Edinburgh EH9 3DF, UK
Francesc Planas-Vilanova
Affiliation:
Departament de Matemàtiques, Universitat Politècnica de Catalunya, Diagonal 647, ETSEIB, E-08028 Barcelona, Spain (francesc.planas@upc.edu; bernat.plans@upc.edu)
Bernat Plans
Affiliation:
Departament de Matemàtiques, Universitat Politècnica de Catalunya, Diagonal 647, ETSEIB, E-08028 Barcelona, Spain (francesc.planas@upc.edu; bernat.plans@upc.edu)
*
*Corresponding author.
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Abstract

Let A ⊂ B be an integral ring extension of integral domains with fields of fractions K and L, respectively. The integral degree of A ⊂ B, denoted by dA(B), is defined as the supremum of the degrees of minimal integral equations of elements of B over A. It is an invariant that lies in between dK(L) and μA(B), the minimal number of generators of the A-module B. Our purpose is to study this invariant. We prove that it is sub-multiplicative and upper-semicontinuous in the following three cases: if A ⊂ B is simple; if A ⊂ B is projective and finite and K ⊂ L is a simple algebraic field extension; or if A is integrally closed. Furthermore, d is upper-semicontinuous if A is noetherian of dimension 1 and with finite integral closure. In general, however, d is neither sub-multiplicative nor upper-semicontinuous.

Keywords

integral extensionintegral degreeintegral closurefield extensionDedekind domainNagata ring

MSC classification

Primary: 13B21: Integral dependence; going up, going down
Secondary: 13B22: Integral closure of rings and ideals ; integrally closed rings, related rings (Japanese, etc.) 13G05: Integral domains 12F05: Algebraic extensions
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 1 , February 2019 , pp. 25 - 46
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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